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e=4x-Q -t-2jQ- OX- -t 9 - =- b

~------------------

07-- L

b1 L

(

-- - _- - ---- - -- - - ~ ----- shy---~_ ---~----~ ----_---- -- -shy

O-ltX ) 0 ~

--_-----------------=- -4 -shy

1~ F- Jr- --~~ = 1shy(l- b2shy_ ~) J

FAt L212) e tlllt) (

1

bull__ bull bull --- -__ -_ -~----- _--_------_- ____-----

C ~ (tH )

F ~ (1J)

C0

t)

8 ---9 t ~t

- -

(~)

W Pig (~~~) f2~~)i~ f (9 I r

of = bQs~ -t ~~~ ~Gl-)

~~~~~~~~

-~~-~-- ~ _-- _-~--

amp (() I bG8~9J ~

lCP Q ~ a~fle )

(J1 61-t- =-t~prL lcamp1L

- ~J2 + a- 2 = 2- (Ji-tl ) ----shy

fbl

12+1

-J2 -

L-h-l)

wY~~_ __ _____ _ _ _Lshy - --shy

ltl _ l h ~- -J

ps (~~(j ~)T~~ cd middot~middot~l~~t~middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoti) x~_ + j~~~ ~ltr-

0 b

bull 9

Q=S

~~- ~O~ ~ -c-~j~

~~gt g (~ ~ go ~ h

bkCbsamp = oh ~ - LD L 1__

L -+~ (xIC~ e b~-T~~~t)~lcrjl~( 1Clmiddot

D~J n CLK--~JCT - ~

os-o- Pty -I T ) ~

~ fi traquoe- ~

t~ at ~yjpm

aQrt- - ~-= 0-61 -)lt jl

bQt _ 2 CL~- bfl-

~2 Q-

~ lt bull bull

-

~ -

~-~ ~gt -j

tf1r ei Z Of

lt

- ~~

-j-

(

3L - 2-~ - J

T J~ =- Ln-

------__-~~-_------ _--shy

(16) ~ P (Go l~~~) ~~)

ctD L - ~ ~ -kane~~

V2pmiddotmiddotmiddotmiddot

~tG- -- 2shy-3

~~~7middoth

_ J- ~

eta3~et)

AB=-3ur) P

h)

(

~ 1-

---- -- -- -middotmiddot----middotmiddotmiddot---------middot--middot-----r------~--middot------(Tb~ ) ~~

CJB) ---reg~ ~ I f 4- rfmiddot (l raquolt

----------- _--- --------__------- ---shy------~-i---

xQ -t 1 z LCl2J CampJ

(b~ eog1K

~~

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-nrlshyoJ- __

~ --shy ~~~nl0T~ ~ Qs (lS-+lC4 )

~

(

fJo-l~ 1 + amprnLbL

2- h1--- 1-h lshy

-- --- ---- -_

~

wt~J) C0

90

0_ PR =)()

~0 (

~) CL) ~

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_ (JL-o ~ -shy

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cD-~

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Sb P(jJ- if L-yen -6J

CS tUl

CO ~ R- tJ)lli~ I b~) kd

-h ~ oll-tCcSJ9J lshy

Lshy

(

r

( G -t) 2- -+ 1 L 7-

1)2-shy

~

2-~ c -X- ~ Lt

~+rx- -yenLtp-O

~ -

f ( srD J (i~4i~jmiddot ~

Nt- ~-~rd- i~6~ ~ J

2 h ~ Q 2L = b _t~(9 - ~tr)~middot

irtn~ -

1- --- __---- ---- _- ----- ~~--------

-~~~F~tmiddot - (

) 0tlshy 0

~ (8 sDmiddot So ~~ is

l j(- ~2- -tshy 1 L _=-=~= - -L

6t1 4fQ

bL ~ a-Q Ll--cLj

bl b4l~)=- Ltb

~===~-_ __ _----- ======----shy

----~-------_-------------

g 2- l 2) - -t- 4 - 9 L Oi -

9t2 (2- ) =-- - ~ y - s L 0 -

lp kJ m~ -c ~ ~ Emiddot

(

--~

p ---~

~

lt ffO~

~

( shy

_ r -~

gt

Ib

~ - =1---~ + Lt) ~

lk~~~l (

j shy 11YXshy -1 [lQ+b~Lr b 1-

j ~ mIL plusmn ri1nt~I~tj

cr middotmiddot(L

-pf f+

e p ~

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~ _J

tJ~ j

~

t+

9 ]Vr---

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+ ~

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cr ~

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t~

v o0

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t ~ ampb -t

~bull i -lib- ~

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r- ~

j

cp ~

~

~

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CD

( - YJ3 -~) [3 3gt

JC -shy t 5l f -1JI

Ql) (6 C- -- bS3 f3gt

( gt

( ~

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-i-l~~J 2 0 46

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Q

(- ~

s

j O~ eqr~~tFl1~~~b __c~_~-

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-m ~_L tl- J

(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

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r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

_

lshy

1 IttP

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I~II~

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(

T

7) d j ) + 1-1 L

2shy

(

Mshy

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( ~~ 01shy ~~~

0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

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o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

e- I0 6 [)~i(P-UJ Cort - Cr(amp -fJ -1~ o]

~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

I yr~) Ci( (~ z I

ViJ-e plusmn = a

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~ -- ArC

( Il v ---t~ = ~ 01f)

f (-i~~)

ampSJ J--~~ ~~

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u )lt L L

T~+ =zt

e) (~) ~ - c= i(yen~ ) _ (

J( z cit fLe- rLf~ ~( ~ e ~

tgt poundL- it

~ - 4-- +c9 - + rL -7 ~ c~ - cshy

0) Co-ho-= i -( s z- h=- plusmn+(~

Al~ 19 ~ $ +lt ~~ = -f if t I

(

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-i+f-~1

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Ji-fl~f Cba b)

(

~ )Itl t L - 1--_ - by + YL--~

aLshy

1~J A ~ o __4y 1 kl---lt 4~

(I0 -- 4-y) 1---- L(( L L

~ ~ ~

o-shy

qD 2 _ -+~ 1-( (cJJ _ G) = o

A -( LI a _ Iuuct + ~-(oo =-0

0

) (a- b ~ (q ~ LV) 0

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r

Of f

(

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j I euroj-J~j ~ Y~ -4JL rTr1+ 2 -4x t ~

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4 r 17 l--~ r

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c

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~r ~~y v ~((II 11)11

~ k ) 1 ~ (JfJ )~ c

crf~J C-M--f1

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- l L

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[t (v7 LOOaI l-5~~ )

Ivvu r Jl ik-- t I

(

raquor- ~L 1

~ qr~ t

(J r~ ~ ~ (~GII fP ( 1-5(~ j

h 60- ~~ Y 2-11gtV91

( ~ (~Jlt)~ ~ V(~ J ([a0

Y

shy

) ~ sect 51~(t7 7 LD lt) (t h~ 9- K -) 1ampt () gt plusmn fT-middot

Ji~ =- I J- ~l V t )

~ - AJD-

+ y2S0vpound 1

yL-shy~ J~sect3lk

LAv~ poundclt ~ 3- C6j 1--lt _ 1 fco+-~l ~~

gt 4- tofY IS fh l( lshyit

) iJ~L iA ~) 1~L( =j I

( (

)~ 2-lt 0 Y~r ~t r if I yen ) 0lt ~ zt-((1--- M~ ~((l--( ltt

~ - Al~

(

-) -fea)--z

~kJ -7 (t lI ~ -= sectr~(~ rNJ)cgf-e ~ 1 SU-lt

o~~~-k ~ fc wl v~f ~ c [Csez~ ~ (~tZ J- f r2

~) ~ gt~ Lv~ ~~ c lt2 0 -k ltlt - 1 r Y L~ 1

- I)( 1~ J plusmn- v o1-t-q gtL

1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

iJ-J~-TrrgtJA (~I~)

c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

) -- shy-~I~-

( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

(

l7 cent e l -

~)olt ) I_qshy

1e Jshy

lt (

) I0c shy 4shy 7 -) q L gt ( 4 ft 0)

o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

1- C(J~ ~ Y- flh~ ~ 1A- or )shy

UICvlgtlt-gt1tid (+(1-) ~

L~ 51 rf- A~~ f r 1

(2- i t- ( f ~ 7~ - i~

7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

-) p~ ift ~ It lt----- or ~ -r

-) 0deg1 f ~ AjL

(

_ ( L5f - ( 1

T~)L+ ~) L C

tIffiI

~ rmiddot- ~ ~ tf7~ ~ T

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-( ( verY ~ -- q 1 V 60 1G -t

q

I

( c1r J r-C A (

r - T I - v- 1

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~ ltL ~~ i~ r ~ t rlr~~ c1Vo ~ k~ SJd~u ~ 1a

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cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

pound ~ fJoQ ~~ (V CO IU( 2- f I iJ1-~ 1 AL cent - Y - I

i (~cent~ - y~ (u _LJ -1 ~~ Llt)9shy

~ V5 ~ (

~[-ch)( shy(~I +00 ~~

tL t __ ~ vr

7) -1 (20-- ~~ (1--6sLJ ) gt

p 4- c a Jb + iffLltgtrJ- l (r- C ) J~J- 0

) C -bU 0 f 7) 0 xG- 111L (

f 0

( o r

r 4-(0)19 -f- IIgt - )Le-f4 s ~

iJ LofO Ifplusmn~ 2J)s Jxi

f r- ~

)

-+~-~(-J

~( - 1 r (i(~o) J-shy L

~

I

etto I)

) L -+ y 1-_ I

(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

--1deg1 I feJ-q ~ to 1to (()) ~ljllLlvJ7htshy+ ~J Q lt= ~ ~ (fyengt-)

WmiddotI-c oc o - 1plusmn j t

- -

) r l

u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

- JJ

~I L

(j~l -t ~i~1 ( ) r-r --1 - ~U(

-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

-shy

-) C( ~~2i b ~ f--J

11 1Jamp 71 Ovb z J iT1 ~ 3

~~ FY 6 0Lu cJ t ~ 1-l-Y = b~

m~ v~ == ~

-- at ~ 1 +y~ co)

~-S-~_DT~O bull -~s-~+ff~

( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

~ l~ pnr -lNX Nw~middot oj 0 r~

f hiUUA 1N- d ~ p oJshy

~~ ~ Am Lb)

- --

~

~ ~

f ~ n

~o

re ~~ ~ I P tJU ~~ J Y --0

tf 1 ~ r t ~ ~

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J t ~

y 5-~ ~ ~ )J ~ 1

rgt l~ t--- ~ craquo ~ gtJ - 11 ~ ir shy

~ 11 ~1 jI

i ~

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J I - + 0 ~r +9

0 - 1 LYV t ~

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~ r-laquo r v1

1t jJ ~ )gt 11 t ~ g

J JJ ~

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(p -J

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~-~

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l k1 ~ h elii~

~ (Jrc l1rrT ~ ~

lj 2shy3m + ~m ~l ~O

( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

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~ 1- L + ~) 1 (b )J- ~

z ~b ~ hrVgt (A)

~ YO f p~ ~ E

AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

U 2shy-z) ~- e f

~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

- --

N A-rR))( M A--TLJ--

~Wamp( b ~d

~ + ~~ -o-- pl-shy

cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

-gt ~ -z ~ -t S4 10 )- 0 l 10 gtshy

cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

LotLU ~ ~meh-~middot

w ~Y etmiddot ~ v

- -

~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

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if9

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~)= ~-- 1 - e2- C c~

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e -- -) -2shy

( c) ~f2- k~ ~

D) ~ f ~ ZoJ-- a 2-b- Q

et--Q =- ~ ~ ~ ~ bullI

a2--e 1z (1 ~ ~ 2-- tL t - 2 C

(OJ I~rt--

-t

tf

~ ~--C

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i ~ ~

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+

~ I

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~ 5

l)h cstr bullbull~

(

1 ~ ~ Cpound Q) 1Jgtl + ~ ( tl i-~~-~middotf _~ (~_ M ~Q

1

- 1

rp ~

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~ 1 t l

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Yr I I rl~ (

1 - -gtr IJ~~1 ~ tr 0 r i

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camp (

1shy-4

r

bull

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gt- to ~ -(

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(

aL -

(

Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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-shy~)- ~

L -

Qshyl- bL shy~ ( 1gt- shy

~ ~ - L

) ~ -~-

~~---l lt bull lmiddot~ u

)1--bull~~ ~~~bull( -]3 shy

-X(lt--

i

-

~~~l~ ~~~ft

--=--~==========-ft1rE3~ ~=-

j-4-~ ~tIl-~

~=~

~rC

~ ~~ ~ ~~J ~

)b

)

4shy

-t C g

c ~ ~~OuI

(

ltj l yengt) - 1--) ~ ) - ~ ( llshy - t1 --+ ) ~ t- I)~

3 C) - )shy +shy ~ C~- t-)shy ~ J-~

( R~~L ~ l-)l -

5 9

reg

~ A T C -t ~ - ~ 0 - s ~~

( YoT J--~ - ) i- ~ ( d - IJo l-) ~- l r~

r--shy

0oc - JL---shy

L~

l shy -

(

-~----------~ ~----~~~

- _w ____ -== __shy-~~-==

t J(~~lL +lt1 - ~-_

~

(

J -- -- ~- Jb 2gt shy

~ ~

regshy

1~ (~ --Jshy + Ltj (a-~)L ~

~ t

l shy L))shy shy (rl _ ~)L shy

~

~ 1

J- 0

- (~~ cL~)

1

(

regshy

I

~ ~~~~-~ ~~) ~ ~b ~

-C 1-~)

(~~~ +~ - ~l-_ ) ~

( t~-)

~ ~1gt ~~~ ~~ ~

~(~-)T ~ ~llti~) ~- c~(~)

~ Cb(~)

~C~)

l ~~~~ ~~)~

shy

--- - ~~~~~ ~ ~ Cogt L

4- ~Y ~V ~rlt ~l- -~

~ (lshy

r(s~2 1 ~-~) ~ ~~~)~~ ~s~~

__ ======~=======~---__shy_---

middot-_ _ --shy

c $-b

i

( shy

Q- Cl ~ ) ~S~

( ) ~ l~t~~) ~~Q

~

~-_ f- bullt C)( ~-

5~f~ -shyuJ)v ~

__ - - _ _ _-_ ___ _ - __ _ - _ __ _ reg

_ (

(

-rr4

(

~ ~

q~

_~---__-__-----~~_ - _ _- _ _--_ _

r-_ ~1

~ ~~- bf~ - ~(~) -- ~~C~) ~ ~(~~ o ~ 1

( c ))

~ ~ f~ ~~

(

---

~~) ~t~1 ~~~ ~

t laquo rI ~u JJ f I~

- S~Y ~l- ~ 0( i)v

t ~u~middotmiddotmiddotmiddotcJ ~~~ ~

--

GD- ~~~ ~~

~~ ~

t~

~shyQfshy

t( i ~ - (~ li)0

-~J~

yshy(

t

oL ~-- ~ -L o )7- C) l tgt

rmiddotmiddot

raquo

a i - 0- e

l OL 0~ 01 - o-Q 1 -= shy

h ~_ ~ ~e~~~ ~

( ~ - J

~ 7 ~ ) h ~1~ tl- v -

-~ ~~~~~- ~ ~ th I d- ~ Jgt~ ~~

~

__----==- _ __ o-c=oco~--middot= = _ -__ _ __ ___~ ==--=__== Jo1 rfI~t ~

~~ et~

- _4- ) + 3L

3

- 4- ~ -t-~-~

3 ( -t- 4- 1 ~ )-1

~ ~ ~T-~) ~ --~ l fZgt1-- )

~-- )-~ ~~~i

~ aL- ~ Qk

- ~~ ~ ~ (lpS

( G ~ ~

CC~ - ~)- ~~ ~(1( ~ ~ - L

) 06 ~~-~~ )

- l~~)

~-~ 1shytCl- ~ ) ~

(~~-) ~

- L -IIL 1 ~ L - 0- 1- Va 1- 1- ~

1 (L-_~) 1- l(tli- ~-) - L

- L~

~ t - ~

middot 1 L bull1 h~

(

)--~t~ -~ ~ ~~

~ r-_ (~-~)

~ NVD ~ ~~ Y-

)0-3 11shy --T ~ 1 -- s =3 ~CJL shy ~ shy t ~ ~

bo 0 - ~ - shy -) ~ amph~ - ) ~~ ~

_--_ _

r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

I~ -- ~ L - ~ 1shy

d ~ YrI1 ~ ~ ~ VY1 t- L

~ ~~ 1h6)+ C ~)

~ ( - roo rt )- - gtV1- 1- L

) -) ~l- _ V1 - r ~ tI 4- l-

~ k 9 ~ -egt~ laquo (f = lshy J~ -0

I ~

~ ~k ~~

R ~ n~)L ~

~ ~- -~-)

~~ ~v~-)

lC1-rtL l b)

- lP-~ ~ l~L - ~-) -~

yenl ~n L ~~~ ~( ~ ) ~ ~ K

bull__ bull bull --- -__ -_ -~----- _--_------_- ____-----

C ~ (tH )

F ~ (1J)

C0

t)

8 ---9 t ~t

- -

(~)

W Pig (~~~) f2~~)i~ f (9 I r

of = bQs~ -t ~~~ ~Gl-)

~~~~~~~~

-~~-~-- ~ _-- _-~--

amp (() I bG8~9J ~

lCP Q ~ a~fle )

(J1 61-t- =-t~prL lcamp1L

- ~J2 + a- 2 = 2- (Ji-tl ) ----shy

fbl

12+1

-J2 -

L-h-l)

wY~~_ __ _____ _ _ _Lshy - --shy

ltl _ l h ~- -J

ps (~~(j ~)T~~ cd middot~middot~l~~t~middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoti) x~_ + j~~~ ~ltr-

0 b

bull 9

Q=S

~~- ~O~ ~ -c-~j~

~~gt g (~ ~ go ~ h

bkCbsamp = oh ~ - LD L 1__

L -+~ (xIC~ e b~-T~~~t)~lcrjl~( 1Clmiddot

D~J n CLK--~JCT - ~

os-o- Pty -I T ) ~

~ fi traquoe- ~

t~ at ~yjpm

aQrt- - ~-= 0-61 -)lt jl

bQt _ 2 CL~- bfl-

~2 Q-

~ lt bull bull

-

~ -

~-~ ~gt -j

tf1r ei Z Of

lt

- ~~

-j-

(

3L - 2-~ - J

T J~ =- Ln-

------__-~~-_------ _--shy

(16) ~ P (Go l~~~) ~~)

ctD L - ~ ~ -kane~~

V2pmiddotmiddotmiddotmiddot

~tG- -- 2shy-3

~~~7middoth

_ J- ~

eta3~et)

AB=-3ur) P

h)

(

~ 1-

---- -- -- -middotmiddot----middotmiddotmiddot---------middot--middot-----r------~--middot------(Tb~ ) ~~

CJB) ---reg~ ~ I f 4- rfmiddot (l raquolt

----------- _--- --------__------- ---shy------~-i---

xQ -t 1 z LCl2J CampJ

(b~ eog1K

~~

QLmiddot- I n+~tmiddot_middot~o rt lt r6)middotKImiddot -+ ~4) + amp bull-t ~- amps_K ~ ~-_- UD~~ ---t-~Il V_ a ) t shy1- --J

-nrlshyoJ- __

~ --shy ~~~nl0T~ ~ Qs (lS-+lC4 )

~

(

fJo-l~ 1 + amprnLbL

2- h1--- 1-h lshy

-- --- ---- -_

~

wt~J) C0

90

0_ PR =)()

~0 (

~) CL) ~

-~ - --gt ~ bull -shy -

_ (JL-o ~ -shy

~ - h-lt-h -l shy~ ~

cD-~

~(IJ vl-- -b~ ---yl

Sb P(jJ- if L-yen -6J

CS tUl

CO ~ R- tJ)lli~ I b~) kd

-h ~ oll-tCcSJ9J lshy

Lshy

(

r

( G -t) 2- -+ 1 L 7-

1)2-shy

~

2-~ c -X- ~ Lt

~+rx- -yenLtp-O

~ -

f ( srD J (i~4i~jmiddot ~

Nt- ~-~rd- i~6~ ~ J

2 h ~ Q 2L = b _t~(9 - ~tr)~middot

irtn~ -

1- --- __---- ---- _- ----- ~~--------

-~~~F~tmiddot - (

) 0tlshy 0

~ (8 sDmiddot So ~~ is

l j(- ~2- -tshy 1 L _=-=~= - -L

6t1 4fQ

bL ~ a-Q Ll--cLj

bl b4l~)=- Ltb

~===~-_ __ _----- ======----shy

----~-------_-------------

g 2- l 2) - -t- 4 - 9 L Oi -

9t2 (2- ) =-- - ~ y - s L 0 -

lp kJ m~ -c ~ ~ Emiddot

(

--~

p ---~

~

lt ffO~

~

( shy

_ r -~

gt

Ib

~ - =1---~ + Lt) ~

lk~~~l (

j shy 11YXshy -1 [lQ+b~Lr b 1-

j ~ mIL plusmn ri1nt~I~tj

cr middotmiddot(L

-pf f+

e p ~

+shyCT ~

+(

~ _J

tJ~ j

~

t+

9 ]Vr---

~Pr ~

+ ~

-t~

cr ~

_

t~

v o0

~

- )bull1 middotmiddotmiddot~fr~r C bull _

t ~ ampb -t

~bull i -lib- ~

~ ~

r- ~

j

cp ~

~

~

~ ~

CD

( - YJ3 -~) [3 3gt

JC -shy t 5l f -1JI

Ql) (6 C- -- bS3 f3gt

( gt

( ~

~L ~ amp~lpoundf1~ ~ b~ - 3bCiise ~ y

-i-l~~J 2 0 46

~b~ _ 0 -- b lt

Q

(- ~

s

j O~ eqr~~tFl1~~~b __c~_~-

~- ---gt B~~IA) ( ctOJo)gt -

-m ~_L tl- J

(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

~~ of ~(Mrt ~ tl~ F~ bu~ t

~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

_

lshy

1 IttP

_~ 1 10-lJV

( C0)9shy~

JJ1shy-- - I+2shy I

I~II~

- =ULshy

(

T

7) d j ) + 1-1 L

2shy

(

Mshy

-11 J~t~) ~

( ~~ 01shy ~~~

0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

~ Vt~ -Lp o~ ~+flt Ik-- ~11c

~c f fGfshy

~~(+ 1A- ~ru f~ fCed t Y~-J(R-o) ~o

IL fnJ u~ lk~ ~~~~~ ~cf(l- ~ lt) C~r~ cf ~-k~J (

~~ Y 0+-) shy~ J rO)tJ~ BY)-~e- poundLlr aP---- ~

f ( O-Lc-W I ~~tL9) (( 0 c )~ f ~ J (gt -it (oreUJ I

o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

e- I0 6 [)~i(P-UJ Cort - Cr(amp -fJ -1~ o]

~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

I yr~) Ci( (~ z I

ViJ-e plusmn = a

r(~ CR~c-r f~~ (0 (C) 0 r (0 (-oj

~ -- ArC

( Il v ---t~ = ~ 01f)

f (-i~~)

ampSJ J--~~ ~~

-(9 (

~vM J (+go) e (~-1(o)

~~ J ~ -fJrvtf

u )lt L L

T~+ =zt

e) (~) ~ - c= i(yen~ ) _ (

J( z cit fLe- rLf~ ~( ~ e ~

tgt poundL- it

~ - 4-- +c9 - + rL -7 ~ c~ - cshy

0) Co-ho-= i -( s z- h=- plusmn+(~

Al~ 19 ~ $ +lt ~~ = -f if t I

(

)IL Q

-i+f-~1

]~ QV)~ ftt u~ ( q-_ r) ~

Ji-fl~f Cba b)

(

~ )Itl t L - 1--_ - by + YL--~

aLshy

1~J A ~ o __4y 1 kl---lt 4~

(I0 -- 4-y) 1---- L(( L L

~ ~ ~

o-shy

qD 2 _ -+~ 1-( (cJJ _ G) = o

A -( LI a _ Iuuct + ~-(oo =-0

0

) (a- b ~ (q ~ LV) 0

-) Q1_ l -ashy 1Jo f Q -- 2 cJ

r

Of f

(

t k yfl-=zl 1

eJlJ ~trJ J ) t r-~ ~ y ~ ~r Jro-alt-I bshy

j I euroj-J~j ~ Y~ -4JL rTr1+ 2 -4x t ~

~ J I Y-+ 4gt i+--v

+N - Hshy

4 r 17 l--~ r

-~_t1

tl

lwt ~ 0 ~J11 yo -~ ]I j Jaf)l~)+ Oit) ~---~~--

~i~ ~ V~ ~(~+l J 4 J 1- =I (~( ~ ) t

c

0 -t it ~ z -+- gt~

(4- x~)lt CJ ~J ~ I

S T ~ (+-1 I1 )

ampf~~ -JrJ~ 7lt0 - gt j ~O-yl T ~)

~ - ~ ~ (11) (

~gt C4y~~~) ~ f S-

~r ~~y v ~((II 11)11

~ k ) 1 ~ (JfJ )~ c

crf~J C-M--f1

~I ~~~) -+ 2r _-+ s AT tV

Yr =- plusmn (YI( 1

- l L

T~ f ~r

-p-t ( p-lshybe- (0 Ca)Xgt tM~

[t (v7 LOOaI l-5~~ )

Ivvu r Jl ik-- t I

(

raquor- ~L 1

~ qr~ t

(J r~ ~ ~ (~GII fP ( 1-5(~ j

h 60- ~~ Y 2-11gtV91

( ~ (~Jlt)~ ~ V(~ J ([a0

Y

shy

) ~ sect 51~(t7 7 LD lt) (t h~ 9- K -) 1ampt () gt plusmn fT-middot

Ji~ =- I J- ~l V t )

~ - AJD-

+ y2S0vpound 1

yL-shy~ J~sect3lk

LAv~ poundclt ~ 3- C6j 1--lt _ 1 fco+-~l ~~

gt 4- tofY IS fh l( lshyit

) iJ~L iA ~) 1~L( =j I

( (

)~ 2-lt 0 Y~r ~t r if I yen ) 0lt ~ zt-((1--- M~ ~((l--( ltt

~ - Al~

(

-) -fea)--z

~kJ -7 (t lI ~ -= sectr~(~ rNJ)cgf-e ~ 1 SU-lt

o~~~-k ~ fc wl v~f ~ c [Csez~ ~ (~tZ J- f r2

~) ~ gt~ Lv~ ~~ c lt2 0 -k ltlt - 1 r Y L~ 1

- I)( 1~ J plusmn- v o1-t-q gtL

1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

iJ-J~-TrrgtJA (~I~)

c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

) -- shy-~I~-

( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

(

l7 cent e l -

~)olt ) I_qshy

1e Jshy

lt (

) I0c shy 4shy 7 -) q L gt ( 4 ft 0)

o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

1- C(J~ ~ Y- flh~ ~ 1A- or )shy

UICvlgtlt-gt1tid (+(1-) ~

L~ 51 rf- A~~ f r 1

(2- i t- ( f ~ 7~ - i~

7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

-) p~ ift ~ It lt----- or ~ -r

-) 0deg1 f ~ AjL

(

_ ( L5f - ( 1

T~)L+ ~) L C

tIffiI

~ rmiddot- ~ ~ tf7~ ~ T

-) r laquo) y -r- tG2sf - ~ ~ o

-( ( verY ~ -- q 1 V 60 1G -t

q

I

( c1r J r-C A (

r - T I - v- 1

~ -- (ArD)

~ ltL ~~ i~ r ~ t rlr~~ c1Vo ~ k~ SJd~u ~ 1a

~) 1 t~-t ~L f- J2Lf1--pound~tL z 2Q ~ q z I~

cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

pound ~ fJoQ ~~ (V CO IU( 2- f I iJ1-~ 1 AL cent - Y - I

i (~cent~ - y~ (u _LJ -1 ~~ Llt)9shy

~ V5 ~ (

~[-ch)( shy(~I +00 ~~

tL t __ ~ vr

7) -1 (20-- ~~ (1--6sLJ ) gt

p 4- c a Jb + iffLltgtrJ- l (r- C ) J~J- 0

) C -bU 0 f 7) 0 xG- 111L (

f 0

( o r

r 4-(0)19 -f- IIgt - )Le-f4 s ~

iJ LofO Ifplusmn~ 2J)s Jxi

f r- ~

)

-+~-~(-J

~( - 1 r (i(~o) J-shy L

~

I

etto I)

) L -+ y 1-_ I

(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

--1deg1 I feJ-q ~ to 1to (()) ~ljllLlvJ7htshy+ ~J Q lt= ~ ~ (fyengt-)

WmiddotI-c oc o - 1plusmn j t

- -

) r l

u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

- JJ

~I L

(j~l -t ~i~1 ( ) r-r --1 - ~U(

-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

-shy

-) C( ~~2i b ~ f--J

11 1Jamp 71 Ovb z J iT1 ~ 3

~~ FY 6 0Lu cJ t ~ 1-l-Y = b~

m~ v~ == ~

-- at ~ 1 +y~ co)

~-S-~_DT~O bull -~s-~+ff~

( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

~ l~ pnr -lNX Nw~middot oj 0 r~

f hiUUA 1N- d ~ p oJshy

~~ ~ Am Lb)

- --

~

~ ~

f ~ n

~o

re ~~ ~ I P tJU ~~ J Y --0

tf 1 ~ r t ~ ~

~ rgt~f 11

J t ~

y 5-~ ~ ~ )J ~ 1

rgt l~ t--- ~ craquo ~ gtJ - 11 ~ ir shy

~ 11 ~1 jI

i ~

1 tv-- ~ t ~

J I - + 0 ~r +9

0 - 1 LYV t ~

IT ~~~ ~ CO ~~~

~ r-laquo r v1

1t jJ ~ )gt 11 t ~ g

J JJ ~

-- middot~1~ ~ ~

~ r (gt

(p -J

~ ~ ----

L--S- ltraquo~~ ~

~~ t

~-~

~~ ~ 1- f-h -~~-------I~-r---

l k1 ~ h elii~

~ (Jrc l1rrT ~ ~

lj 2shy3m + ~m ~l ~O

( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

~~ ~ c t-7(g t~ +PI P2) d-BMu

~ 1- L + ~) 1 (b )J- ~

z ~b ~ hrVgt (A)

~ YO f p~ ~ E

AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

U 2shy-z) ~- e f

~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

- --

N A-rR))( M A--TLJ--

~Wamp( b ~d

~ + ~~ -o-- pl-shy

cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

-gt ~ -z ~ -t S4 10 )- 0 l 10 gtshy

cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

LotLU ~ ~meh-~middot

w ~Y etmiddot ~ v

- -

~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

~ - l 0 ()-Z7- + ----) ~ --- + ~ ==-1 0)- 10)- ()L kshy

-) 0- ~ ~ r bi--- ~

__ ~ S--

if9

(

0 I-e_

cJJishy middot 2- 102-)t e-~ -17] ~~

~)= ~-- 1 - e2- C c~

a)shy

e -- -) -2shy

( c) ~f2- k~ ~

D) ~ f ~ ZoJ-- a 2-b- Q

et--Q =- ~ ~ ~ ~ bullI

a2--e 1z (1 ~ ~ 2-- tL t - 2 C

(OJ I~rt--

-t

tf

~ ~--C

~~ t-

i ~ ~

~ ~ r

+

~ I

)+ ~

~

-s i

~1-

~ poundshy g~ i ~

__~~

i ) ~~ ~ ~ ~ oot ushy

~ 5

l)h cstr bullbull~

(

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1

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r

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aL -

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Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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ps (~~(j ~)T~~ cd middot~middot~l~~t~middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddoti) x~_ + j~~~ ~ltr-

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CJB) ---reg~ ~ I f 4- rfmiddot (l raquolt

----------- _--- --------__------- ---shy------~-i---

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(b~ eog1K

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Lshy

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r

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f ( srD J (i~4i~jmiddot ~

Nt- ~-~rd- i~6~ ~ J

2 h ~ Q 2L = b _t~(9 - ~tr)~middot

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1- --- __---- ---- _- ----- ~~--------

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) 0tlshy 0

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l j(- ~2- -tshy 1 L _=-=~= - -L

6t1 4fQ

bL ~ a-Q Ll--cLj

bl b4l~)=- Ltb

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----~-------_-------------

g 2- l 2) - -t- 4 - 9 L Oi -

9t2 (2- ) =-- - ~ y - s L 0 -

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t ~ ampb -t

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Q

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j O~ eqr~~tFl1~~~b __c~_~-

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(Uh)

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900 ot ~

---- - ~----------~----- --------------- -------~=---

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r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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~- o~ ftJshy ~

c~r~2 Pr c-reg

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1 IttP

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(

T

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2shy

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Mshy

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0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

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IL fnJ u~ lk~ ~~~~~ ~cf(l- ~ lt) C~r~ cf ~-k~J (

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o ( arltJ)

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--

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( Il v ---t~ = ~ 01f)

f (-i~~)

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u )lt L L

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c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

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1Jl AM~111

oL-f-Cj bL 306~

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1e Jshy

lt (

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ff+ (~) ~4c ~jllcl)

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f r- ~

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~~~~=

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jb U~L ~ t Q M J- ~

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( r ldJl-P) ( ~~-Y

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( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

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l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

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i2A PI E I Egt- P M ~r- u-M

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AS$~RTION ANDRIASON~

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(

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aL -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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CJB) ---reg~ ~ I f 4- rfmiddot (l raquolt

----------- _--- --------__------- ---shy------~-i---

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(b~ eog1K

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f ( srD J (i~4i~jmiddot ~

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2 h ~ Q 2L = b _t~(9 - ~tr)~middot

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) 0tlshy 0

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l j(- ~2- -tshy 1 L _=-=~= - -L

6t1 4fQ

bL ~ a-Q Ll--cLj

bl b4l~)=- Ltb

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g 2- l 2) - -t- 4 - 9 L Oi -

9t2 (2- ) =-- - ~ y - s L 0 -

lp kJ m~ -c ~ ~ Emiddot

(

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Q

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j O~ eqr~~tFl1~~~b __c~_~-

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(Uh)

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900 ot ~

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r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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c~r~2 Pr c-reg

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1 IttP

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(

T

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2shy

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0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

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( Il v ---t~ = ~ 01f)

f (-i~~)

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(I0 -- 4-y) 1---- L(( L L

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oL-f-Cj bL 306~

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cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

pound ~ fJoQ ~~ (V CO IU( 2- f I iJ1-~ 1 AL cent - Y - I

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tL t __ ~ vr

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) C -bU 0 f 7) 0 xG- 111L (

f 0

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r 4-(0)19 -f- IIgt - )Le-f4 s ~

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f r- ~

)

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etto I)

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(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

--1deg1 I feJ-q ~ to 1to (()) ~ljllLlvJ7htshy+ ~J Q lt= ~ ~ (fyengt-)

WmiddotI-c oc o - 1plusmn j t

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L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

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(j~l -t ~i~1 ( ) r-r --1 - ~U(

-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

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11 1Jamp 71 Ovb z J iT1 ~ 3

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m~ v~ == ~

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( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

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a-~ q 2 2( mf~ IJl ~+~~l

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f hiUUA 1N- d ~ p oJshy

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( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

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AS$~RTION ANDRIASON~

( ~-

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J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

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a _a dJO a-eshy

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--

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cry )(2- + f _ 1- o ~ b- b

(

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m K _ lJ1shy- Gl a 1shy

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w ~Y etmiddot ~ v

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1

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r

bull

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I

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tpound- ~ ~ JLi~ Ji

f~d r L~ ~ flshy

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aL -

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Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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QIshy 01-shy ~~

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a i - 0- e

l OL 0~ 01 - o-Q 1 -= shy

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3

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(16) ~ P (Go l~~~) ~~)

ctD L - ~ ~ -kane~~

V2pmiddotmiddotmiddotmiddot

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AB=-3ur) P

h)

(

~ 1-

---- -- -- -middotmiddot----middotmiddotmiddot---------middot--middot-----r------~--middot------(Tb~ ) ~~

CJB) ---reg~ ~ I f 4- rfmiddot (l raquolt

----------- _--- --------__------- ---shy------~-i---

xQ -t 1 z LCl2J CampJ

(b~ eog1K

~~

QLmiddot- I n+~tmiddot_middot~o rt lt r6)middotKImiddot -+ ~4) + amp bull-t ~- amps_K ~ ~-_- UD~~ ---t-~Il V_ a ) t shy1- --J

-nrlshyoJ- __

~ --shy ~~~nl0T~ ~ Qs (lS-+lC4 )

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fJo-l~ 1 + amprnLbL

2- h1--- 1-h lshy

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wt~J) C0

90

0_ PR =)()

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~) CL) ~

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_ (JL-o ~ -shy

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Lshy

(

r

( G -t) 2- -+ 1 L 7-

1)2-shy

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2-~ c -X- ~ Lt

~+rx- -yenLtp-O

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f ( srD J (i~4i~jmiddot ~

Nt- ~-~rd- i~6~ ~ J

2 h ~ Q 2L = b _t~(9 - ~tr)~middot

irtn~ -

1- --- __---- ---- _- ----- ~~--------

-~~~F~tmiddot - (

) 0tlshy 0

~ (8 sDmiddot So ~~ is

l j(- ~2- -tshy 1 L _=-=~= - -L

6t1 4fQ

bL ~ a-Q Ll--cLj

bl b4l~)=- Ltb

~===~-_ __ _----- ======----shy

----~-------_-------------

g 2- l 2) - -t- 4 - 9 L Oi -

9t2 (2- ) =-- - ~ y - s L 0 -

lp kJ m~ -c ~ ~ Emiddot

(

--~

p ---~

~

lt ffO~

~

( shy

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gt

Ib

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j shy 11YXshy -1 [lQ+b~Lr b 1-

j ~ mIL plusmn ri1nt~I~tj

cr middotmiddot(L

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t+

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t ~ ampb -t

~bull i -lib- ~

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r- ~

j

cp ~

~

~

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( - YJ3 -~) [3 3gt

JC -shy t 5l f -1JI

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-i-l~~J 2 0 46

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Q

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s

j O~ eqr~~tFl1~~~b __c~_~-

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-m ~_L tl- J

(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

~~ of ~(Mrt ~ tl~ F~ bu~ t

~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

_

lshy

1 IttP

_~ 1 10-lJV

( C0)9shy~

JJ1shy-- - I+2shy I

I~II~

- =ULshy

(

T

7) d j ) + 1-1 L

2shy

(

Mshy

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( ~~ 01shy ~~~

0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

~ Vt~ -Lp o~ ~+flt Ik-- ~11c

~c f fGfshy

~~(+ 1A- ~ru f~ fCed t Y~-J(R-o) ~o

IL fnJ u~ lk~ ~~~~~ ~cf(l- ~ lt) C~r~ cf ~-k~J (

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f ( O-Lc-W I ~~tL9) (( 0 c )~ f ~ J (gt -it (oreUJ I

o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

e- I0 6 [)~i(P-UJ Cort - Cr(amp -fJ -1~ o]

~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

I yr~) Ci( (~ z I

ViJ-e plusmn = a

r(~ CR~c-r f~~ (0 (C) 0 r (0 (-oj

~ -- ArC

( Il v ---t~ = ~ 01f)

f (-i~~)

ampSJ J--~~ ~~

-(9 (

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~~ J ~ -fJrvtf

u )lt L L

T~+ =zt

e) (~) ~ - c= i(yen~ ) _ (

J( z cit fLe- rLf~ ~( ~ e ~

tgt poundL- it

~ - 4-- +c9 - + rL -7 ~ c~ - cshy

0) Co-ho-= i -( s z- h=- plusmn+(~

Al~ 19 ~ $ +lt ~~ = -f if t I

(

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]~ QV)~ ftt u~ ( q-_ r) ~

Ji-fl~f Cba b)

(

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aLshy

1~J A ~ o __4y 1 kl---lt 4~

(I0 -- 4-y) 1---- L(( L L

~ ~ ~

o-shy

qD 2 _ -+~ 1-( (cJJ _ G) = o

A -( LI a _ Iuuct + ~-(oo =-0

0

) (a- b ~ (q ~ LV) 0

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r

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4 r 17 l--~ r

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Yr =- plusmn (YI( 1

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( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

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lt (

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ff+ (~) ~4c ~jllcl)

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+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

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(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

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11 1Jamp 71 Ovb z J iT1 ~ 3

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( r ldJl-P) ( ~~-Y

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S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

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middot--_-_~-- -_-----_ ------- __----- --shy

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i2A PI E I Egt- P M ~r- u-M

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AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

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9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

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37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

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N A-rR))( M A--TLJ--

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cry )(2- + f _ 1- o ~ b- b

(

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m K _ lJ1shy- Gl a 1shy

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1

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r

bull

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aL -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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-shy- 35

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a i - 0- e

l OL 0~ 01 - o-Q 1 -= shy

h ~_ ~ ~e~~~ ~

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~ 7 ~ ) h ~1~ tl- v -

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3

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r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

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d ~ YrI1 ~ ~ ~ VY1 t- L

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I ~

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---- -- -- -middotmiddot----middotmiddotmiddot---------middot--middot-----r------~--middot------(Tb~ ) ~~

CJB) ---reg~ ~ I f 4- rfmiddot (l raquolt

----------- _--- --------__------- ---shy------~-i---

xQ -t 1 z LCl2J CampJ

(b~ eog1K

~~

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-nrlshyoJ- __

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fJo-l~ 1 + amprnLbL

2- h1--- 1-h lshy

-- --- ---- -_

~

wt~J) C0

90

0_ PR =)()

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~) CL) ~

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_ (JL-o ~ -shy

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CO ~ R- tJ)lli~ I b~) kd

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Lshy

(

r

( G -t) 2- -+ 1 L 7-

1)2-shy

~

2-~ c -X- ~ Lt

~+rx- -yenLtp-O

~ -

f ( srD J (i~4i~jmiddot ~

Nt- ~-~rd- i~6~ ~ J

2 h ~ Q 2L = b _t~(9 - ~tr)~middot

irtn~ -

1- --- __---- ---- _- ----- ~~--------

-~~~F~tmiddot - (

) 0tlshy 0

~ (8 sDmiddot So ~~ is

l j(- ~2- -tshy 1 L _=-=~= - -L

6t1 4fQ

bL ~ a-Q Ll--cLj

bl b4l~)=- Ltb

~===~-_ __ _----- ======----shy

----~-------_-------------

g 2- l 2) - -t- 4 - 9 L Oi -

9t2 (2- ) =-- - ~ y - s L 0 -

lp kJ m~ -c ~ ~ Emiddot

(

--~

p ---~

~

lt ffO~

~

( shy

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gt

Ib

~ - =1---~ + Lt) ~

lk~~~l (

j shy 11YXshy -1 [lQ+b~Lr b 1-

j ~ mIL plusmn ri1nt~I~tj

cr middotmiddot(L

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e p ~

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~ _J

tJ~ j

~

t+

9 ]Vr---

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+ ~

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cr ~

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t~

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- )bull1 middotmiddotmiddot~fr~r C bull _

t ~ ampb -t

~bull i -lib- ~

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r- ~

j

cp ~

~

~

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CD

( - YJ3 -~) [3 3gt

JC -shy t 5l f -1JI

Ql) (6 C- -- bS3 f3gt

( gt

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-i-l~~J 2 0 46

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Q

(- ~

s

j O~ eqr~~tFl1~~~b __c~_~-

~- ---gt B~~IA) ( ctOJo)gt -

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(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

~~ of ~(Mrt ~ tl~ F~ bu~ t

~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

_

lshy

1 IttP

_~ 1 10-lJV

( C0)9shy~

JJ1shy-- - I+2shy I

I~II~

- =ULshy

(

T

7) d j ) + 1-1 L

2shy

(

Mshy

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( ~~ 01shy ~~~

0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

~ Vt~ -Lp o~ ~+flt Ik-- ~11c

~c f fGfshy

~~(+ 1A- ~ru f~ fCed t Y~-J(R-o) ~o

IL fnJ u~ lk~ ~~~~~ ~cf(l- ~ lt) C~r~ cf ~-k~J (

~~ Y 0+-) shy~ J rO)tJ~ BY)-~e- poundLlr aP---- ~

f ( O-Lc-W I ~~tL9) (( 0 c )~ f ~ J (gt -it (oreUJ I

o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

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AS$~RTION ANDRIASON~

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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r

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bull2- Lshy

(

bull bull

I ~

~

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n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

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b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

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_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

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1 IttP

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I~II~

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(

T

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lUI JI -f n~ (j

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JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

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o ( arltJ)

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~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

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( Il v ---t~ = ~ 01f)

f (-i~~)

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(I0 -- 4-y) 1---- L(( L L

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1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

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c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

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QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

(

l7 cent e l -

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1e Jshy

lt (

) I0c shy 4shy 7 -) q L gt ( 4 ft 0)

o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

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7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

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(

_ ( L5f - ( 1

T~)L+ ~) L C

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cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

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) C -bU 0 f 7) 0 xG- 111L (

f 0

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r 4-(0)19 -f- IIgt - )Le-f4 s ~

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f r- ~

)

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I

etto I)

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(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

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WmiddotI-c oc o - 1plusmn j t

- -

) r l

u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

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(

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(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

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m~ v~ == ~

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( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

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f hiUUA 1N- d ~ p oJshy

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( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

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middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

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AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

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~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

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N A-rR))( M A--TLJ--

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cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

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cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

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w ~Y etmiddot ~ v

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~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

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l)h cstr bullbull~

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1

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r

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aL -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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j shy 11YXshy -1 [lQ+b~Lr b 1-

j ~ mIL plusmn ri1nt~I~tj

cr middotmiddot(L

-pf f+

e p ~

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~

t+

9 ]Vr---

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+ ~

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cr ~

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t~

v o0

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t ~ ampb -t

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j

cp ~

~

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( - YJ3 -~) [3 3gt

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Q

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s

j O~ eqr~~tFl1~~~b __c~_~-

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(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

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r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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~- o~ ftJshy ~

c~r~2 Pr c-reg

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lUI JI -f n~ (j

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o ( arltJ)

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f (-i~~)

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QL-t cd) 1--= 4- ~

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oL-f-Cj bL 306~

)

(

l7 cent e l -

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1e Jshy

lt (

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o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

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_ ( L5f - ( 1

T~)L+ ~) L C

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cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

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f 0

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r 4-(0)19 -f- IIgt - )Le-f4 s ~

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f r- ~

)

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WmiddotI-c oc o - 1plusmn j t

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u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

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-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

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m~ v~ == ~

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( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

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f hiUUA 1N- d ~ p oJshy

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l k1 ~ h elii~

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( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

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AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

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~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

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N A-rR))( M A--TLJ--

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cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

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cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

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w ~Y etmiddot ~ v

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~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

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l)h cstr bullbull~

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1

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rp ~

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r

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aL -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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g 2- l 2) - -t- 4 - 9 L Oi -

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(

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cr middotmiddot(L

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r

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bull bull

I ~

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-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

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LY == A-X _ ~ It)

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1e Jshy

lt (

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(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

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jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

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tgt)- shy ~

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poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

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LU- N M (h1lt)

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olaquo k d-2- gt( - 2-4

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middot--_-_~-- -_-----_ ------- __----- --shy

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AS$~RTION ANDRIASON~

( ~-

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J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

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a _a dJO a-eshy

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N A-rR))( M A--TLJ--

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cry )(2- + f _ 1- o ~ b- b

(

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aL -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

I~ -- ~ L - ~ 1shy

d ~ YrI1 ~ ~ ~ VY1 t- L

~ ~~ 1h6)+ C ~)

~ ( - roo rt )- - gtV1- 1- L

) -) ~l- _ V1 - r ~ tI 4- l-

~ k 9 ~ -egt~ laquo (f = lshy J~ -0

I ~

~ ~k ~~

R ~ n~)L ~

~ ~- -~-)

~~ ~v~-)

lC1-rtL l b)

- lP-~ ~ l~L - ~-) -~

yenl ~n L ~~~ ~( ~ ) ~ ~ K

( shy

_ r -~

gt

Ib

~ - =1---~ + Lt) ~

lk~~~l (

j shy 11YXshy -1 [lQ+b~Lr b 1-

j ~ mIL plusmn ri1nt~I~tj

cr middotmiddot(L

-pf f+

e p ~

+shyCT ~

+(

~ _J

tJ~ j

~

t+

9 ]Vr---

~Pr ~

+ ~

-t~

cr ~

_

t~

v o0

~

- )bull1 middotmiddotmiddot~fr~r C bull _

t ~ ampb -t

~bull i -lib- ~

~ ~

r- ~

j

cp ~

~

~

~ ~

CD

( - YJ3 -~) [3 3gt

JC -shy t 5l f -1JI

Ql) (6 C- -- bS3 f3gt

( gt

( ~

~L ~ amp~lpoundf1~ ~ b~ - 3bCiise ~ y

-i-l~~J 2 0 46

~b~ _ 0 -- b lt

Q

(- ~

s

j O~ eqr~~tFl1~~~b __c~_~-

~- ---gt B~~IA) ( ctOJo)gt -

-m ~_L tl- J

(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

~~ of ~(Mrt ~ tl~ F~ bu~ t

~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

_

lshy

1 IttP

_~ 1 10-lJV

( C0)9shy~

JJ1shy-- - I+2shy I

I~II~

- =ULshy

(

T

7) d j ) + 1-1 L

2shy

(

Mshy

-11 J~t~) ~

( ~~ 01shy ~~~

0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

~ Vt~ -Lp o~ ~+flt Ik-- ~11c

~c f fGfshy

~~(+ 1A- ~ru f~ fCed t Y~-J(R-o) ~o

IL fnJ u~ lk~ ~~~~~ ~cf(l- ~ lt) C~r~ cf ~-k~J (

~~ Y 0+-) shy~ J rO)tJ~ BY)-~e- poundLlr aP---- ~

f ( O-Lc-W I ~~tL9) (( 0 c )~ f ~ J (gt -it (oreUJ I

o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

e- I0 6 [)~i(P-UJ Cort - Cr(amp -fJ -1~ o]

~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

I yr~) Ci( (~ z I

ViJ-e plusmn = a

r(~ CR~c-r f~~ (0 (C) 0 r (0 (-oj

~ -- ArC

( Il v ---t~ = ~ 01f)

f (-i~~)

ampSJ J--~~ ~~

-(9 (

~vM J (+go) e (~-1(o)

~~ J ~ -fJrvtf

u )lt L L

T~+ =zt

e) (~) ~ - c= i(yen~ ) _ (

J( z cit fLe- rLf~ ~( ~ e ~

tgt poundL- it

~ - 4-- +c9 - + rL -7 ~ c~ - cshy

0) Co-ho-= i -( s z- h=- plusmn+(~

Al~ 19 ~ $ +lt ~~ = -f if t I

(

)IL Q

-i+f-~1

]~ QV)~ ftt u~ ( q-_ r) ~

Ji-fl~f Cba b)

(

~ )Itl t L - 1--_ - by + YL--~

aLshy

1~J A ~ o __4y 1 kl---lt 4~

(I0 -- 4-y) 1---- L(( L L

~ ~ ~

o-shy

qD 2 _ -+~ 1-( (cJJ _ G) = o

A -( LI a _ Iuuct + ~-(oo =-0

0

) (a- b ~ (q ~ LV) 0

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r

Of f

(

t k yfl-=zl 1

eJlJ ~trJ J ) t r-~ ~ y ~ ~r Jro-alt-I bshy

j I euroj-J~j ~ Y~ -4JL rTr1+ 2 -4x t ~

~ J I Y-+ 4gt i+--v

+N - Hshy

4 r 17 l--~ r

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c

0 -t it ~ z -+- gt~

(4- x~)lt CJ ~J ~ I

S T ~ (+-1 I1 )

ampf~~ -JrJ~ 7lt0 - gt j ~O-yl T ~)

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~gt C4y~~~) ~ f S-

~r ~~y v ~((II 11)11

~ k ) 1 ~ (JfJ )~ c

crf~J C-M--f1

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- l L

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[t (v7 LOOaI l-5~~ )

Ivvu r Jl ik-- t I

(

raquor- ~L 1

~ qr~ t

(J r~ ~ ~ (~GII fP ( 1-5(~ j

h 60- ~~ Y 2-11gtV91

( ~ (~Jlt)~ ~ V(~ J ([a0

Y

shy

) ~ sect 51~(t7 7 LD lt) (t h~ 9- K -) 1ampt () gt plusmn fT-middot

Ji~ =- I J- ~l V t )

~ - AJD-

+ y2S0vpound 1

yL-shy~ J~sect3lk

LAv~ poundclt ~ 3- C6j 1--lt _ 1 fco+-~l ~~

gt 4- tofY IS fh l( lshyit

) iJ~L iA ~) 1~L( =j I

( (

)~ 2-lt 0 Y~r ~t r if I yen ) 0lt ~ zt-((1--- M~ ~((l--( ltt

~ - Al~

(

-) -fea)--z

~kJ -7 (t lI ~ -= sectr~(~ rNJ)cgf-e ~ 1 SU-lt

o~~~-k ~ fc wl v~f ~ c [Csez~ ~ (~tZ J- f r2

~) ~ gt~ Lv~ ~~ c lt2 0 -k ltlt - 1 r Y L~ 1

- I)( 1~ J plusmn- v o1-t-q gtL

1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

iJ-J~-TrrgtJA (~I~)

c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

) -- shy-~I~-

( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

(

l7 cent e l -

~)olt ) I_qshy

1e Jshy

lt (

) I0c shy 4shy 7 -) q L gt ( 4 ft 0)

o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

1- C(J~ ~ Y- flh~ ~ 1A- or )shy

UICvlgtlt-gt1tid (+(1-) ~

L~ 51 rf- A~~ f r 1

(2- i t- ( f ~ 7~ - i~

7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

-) p~ ift ~ It lt----- or ~ -r

-) 0deg1 f ~ AjL

(

_ ( L5f - ( 1

T~)L+ ~) L C

tIffiI

~ rmiddot- ~ ~ tf7~ ~ T

-) r laquo) y -r- tG2sf - ~ ~ o

-( ( verY ~ -- q 1 V 60 1G -t

q

I

( c1r J r-C A (

r - T I - v- 1

~ -- (ArD)

~ ltL ~~ i~ r ~ t rlr~~ c1Vo ~ k~ SJd~u ~ 1a

~) 1 t~-t ~L f- J2Lf1--pound~tL z 2Q ~ q z I~

cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

pound ~ fJoQ ~~ (V CO IU( 2- f I iJ1-~ 1 AL cent - Y - I

i (~cent~ - y~ (u _LJ -1 ~~ Llt)9shy

~ V5 ~ (

~[-ch)( shy(~I +00 ~~

tL t __ ~ vr

7) -1 (20-- ~~ (1--6sLJ ) gt

p 4- c a Jb + iffLltgtrJ- l (r- C ) J~J- 0

) C -bU 0 f 7) 0 xG- 111L (

f 0

( o r

r 4-(0)19 -f- IIgt - )Le-f4 s ~

iJ LofO Ifplusmn~ 2J)s Jxi

f r- ~

)

-+~-~(-J

~( - 1 r (i(~o) J-shy L

~

I

etto I)

) L -+ y 1-_ I

(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

--1deg1 I feJ-q ~ to 1to (()) ~ljllLlvJ7htshy+ ~J Q lt= ~ ~ (fyengt-)

WmiddotI-c oc o - 1plusmn j t

- -

) r l

u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

- JJ

~I L

(j~l -t ~i~1 ( ) r-r --1 - ~U(

-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

-shy

-) C( ~~2i b ~ f--J

11 1Jamp 71 Ovb z J iT1 ~ 3

~~ FY 6 0Lu cJ t ~ 1-l-Y = b~

m~ v~ == ~

-- at ~ 1 +y~ co)

~-S-~_DT~O bull -~s-~+ff~

( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

~ l~ pnr -lNX Nw~middot oj 0 r~

f hiUUA 1N- d ~ p oJshy

~~ ~ Am Lb)

- --

~

~ ~

f ~ n

~o

re ~~ ~ I P tJU ~~ J Y --0

tf 1 ~ r t ~ ~

~ rgt~f 11

J t ~

y 5-~ ~ ~ )J ~ 1

rgt l~ t--- ~ craquo ~ gtJ - 11 ~ ir shy

~ 11 ~1 jI

i ~

1 tv-- ~ t ~

J I - + 0 ~r +9

0 - 1 LYV t ~

IT ~~~ ~ CO ~~~

~ r-laquo r v1

1t jJ ~ )gt 11 t ~ g

J JJ ~

-- middot~1~ ~ ~

~ r (gt

(p -J

~ ~ ----

L--S- ltraquo~~ ~

~~ t

~-~

~~ ~ 1- f-h -~~-------I~-r---

l k1 ~ h elii~

~ (Jrc l1rrT ~ ~

lj 2shy3m + ~m ~l ~O

( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

~~ ~ c t-7(g t~ +PI P2) d-BMu

~ 1- L + ~) 1 (b )J- ~

z ~b ~ hrVgt (A)

~ YO f p~ ~ E

AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

U 2shy-z) ~- e f

~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

- --

N A-rR))( M A--TLJ--

~Wamp( b ~d

~ + ~~ -o-- pl-shy

cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

-gt ~ -z ~ -t S4 10 )- 0 l 10 gtshy

cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

LotLU ~ ~meh-~middot

w ~Y etmiddot ~ v

- -

~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

~ - l 0 ()-Z7- + ----) ~ --- + ~ ==-1 0)- 10)- ()L kshy

-) 0- ~ ~ r bi--- ~

__ ~ S--

if9

(

0 I-e_

cJJishy middot 2- 102-)t e-~ -17] ~~

~)= ~-- 1 - e2- C c~

a)shy

e -- -) -2shy

( c) ~f2- k~ ~

D) ~ f ~ ZoJ-- a 2-b- Q

et--Q =- ~ ~ ~ ~ bullI

a2--e 1z (1 ~ ~ 2-- tL t - 2 C

(OJ I~rt--

-t

tf

~ ~--C

~~ t-

i ~ ~

~ ~ r

+

~ I

)+ ~

~

-s i

~1-

~ poundshy g~ i ~

__~~

i ) ~~ ~ ~ ~ oot ushy

~ 5

l)h cstr bullbull~

(

1 ~ ~ Cpound Q) 1Jgtl + ~ ( tl i-~~-~middotf _~ (~_ M ~Q

1

- 1

rp ~

-shy y

~ 1 t l

Y C 1

Yr I I rl~ (

1 - -gtr IJ~~1 ~ tr 0 r i

~

QI~ t

1

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shy

~ middotu~

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)~L c l~ kgt) - --

camp (

1shy-4

r

bull

~-

(-

~ - - ~-- ~lt

~t+ ~-~ + ~ ~r) ~ -~

--(

~~ f ) - ~f-)--~~

~

-- ~amp~yen-- Jl

I

lti)shy

gt- to ~ -(

- Imiddot~cal --~~ ~

( 4 -- ~

b ~ VS -

tpound- ~ ~ JLi~ Ji

f~d r L~ ~ flshy

~ S+ A ~ f~ _ r J~

~ 1shy--

al~LL-=shy

c middot~ rc shy ~

(

~ -~Q

~VP~

(

~ _c~ C ~jlt1__ ~U-

j~f ~ -shy ~ ~

s~-Q - 3 ~ )

~ I

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(~j~~ ~- -r- shyCd ~-J

(

aL -

(

Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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A~~) tt CIL)-1shy G- ~

o Ioshy

(-04) d l bl-Ja ) ~

QIshy 01-shy ~~

l~ l~)

~ 1

A ~ ~3~ - ~

~ ( ~f~t

~ 2q ~ ltrshyI

ltxt~

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(

l shy

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3

laquo~

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~

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ltt-- ~ t~~~ ~~

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-shy- 35

~(shyI lt0 ~ ~ ~ Q

+) ltS~ -- 51 ~gt Ztmiddot~-~~middotmiddotmiddotmiddotmiddotmiddotmiddot

- 2i 9A~

~ S ~

( ffshy lt bull

J 4ltr~r ~b~~~~ ~

0 j~ l~S ~~ ~~~~)shy

middotle~ -lts --~bull shy

-= - h ttll ~)

)~

0) ~ -shyb)

-shy~)- ~

L -

Qshyl- bL shy~ ( 1gt- shy

~ ~ - L

) ~ -~-

~~---l lt bull lmiddot~ u

)1--bull~~ ~~~bull( -]3 shy

-X(lt--

i

-

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j-4-~ ~tIl-~

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3 C) - )shy +shy ~ C~- t-)shy ~ J-~

( R~~L ~ l-)l -

5 9

reg

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(

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t J(~~lL +lt1 - ~-_

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~

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(

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middot-_ _ --shy

c $-b

i

( shy

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( ) ~ l~t~~) ~~Q

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5~f~ -shyuJ)v ~

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t ~u~middotmiddotmiddotmiddotcJ ~~~ ~

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t

oL ~-- ~ -L o )7- C) l tgt

rmiddotmiddot

raquo

a i - 0- e

l OL 0~ 01 - o-Q 1 -= shy

h ~_ ~ ~e~~~ ~

( ~ - J

~ 7 ~ ) h ~1~ tl- v -

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~

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~~ et~

- _4- ) + 3L

3

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) 06 ~~-~~ )

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(

)--~t~ -~ ~ ~~

~ r-_ (~-~)

~ NVD ~ ~~ Y-

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bo 0 - ~ - shy -) ~ amph~ - ) ~~ ~

_--_ _

r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

I~ -- ~ L - ~ 1shy

d ~ YrI1 ~ ~ ~ VY1 t- L

~ ~~ 1h6)+ C ~)

~ ( - roo rt )- - gtV1- 1- L

) -) ~l- _ V1 - r ~ tI 4- l-

~ k 9 ~ -egt~ laquo (f = lshy J~ -0

I ~

~ ~k ~~

R ~ n~)L ~

~ ~- -~-)

~~ ~v~-)

lC1-rtL l b)

- lP-~ ~ l~L - ~-) -~

yenl ~n L ~~~ ~( ~ ) ~ ~ K

cr middotmiddot(L

-pf f+

e p ~

+shyCT ~

+(

~ _J

tJ~ j

~

t+

9 ]Vr---

~Pr ~

+ ~

-t~

cr ~

_

t~

v o0

~

- )bull1 middotmiddotmiddot~fr~r C bull _

t ~ ampb -t

~bull i -lib- ~

~ ~

r- ~

j

cp ~

~

~

~ ~

CD

( - YJ3 -~) [3 3gt

JC -shy t 5l f -1JI

Ql) (6 C- -- bS3 f3gt

( gt

( ~

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-i-l~~J 2 0 46

~b~ _ 0 -- b lt

Q

(- ~

s

j O~ eqr~~tFl1~~~b __c~_~-

~- ---gt B~~IA) ( ctOJo)gt -

-m ~_L tl- J

(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

~~ of ~(Mrt ~ tl~ F~ bu~ t

~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

_

lshy

1 IttP

_~ 1 10-lJV

( C0)9shy~

JJ1shy-- - I+2shy I

I~II~

- =ULshy

(

T

7) d j ) + 1-1 L

2shy

(

Mshy

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( ~~ 01shy ~~~

0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

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~~(+ 1A- ~ru f~ fCed t Y~-J(R-o) ~o

IL fnJ u~ lk~ ~~~~~ ~cf(l- ~ lt) C~r~ cf ~-k~J (

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f ( O-Lc-W I ~~tL9) (( 0 c )~ f ~ J (gt -it (oreUJ I

o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

e- I0 6 [)~i(P-UJ Cort - Cr(amp -fJ -1~ o]

~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

I yr~) Ci( (~ z I

ViJ-e plusmn = a

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~ -- ArC

( Il v ---t~ = ~ 01f)

f (-i~~)

ampSJ J--~~ ~~

-(9 (

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~~ J ~ -fJrvtf

u )lt L L

T~+ =zt

e) (~) ~ - c= i(yen~ ) _ (

J( z cit fLe- rLf~ ~( ~ e ~

tgt poundL- it

~ - 4-- +c9 - + rL -7 ~ c~ - cshy

0) Co-ho-= i -( s z- h=- plusmn+(~

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(

)IL Q

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]~ QV)~ ftt u~ ( q-_ r) ~

Ji-fl~f Cba b)

(

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1~J A ~ o __4y 1 kl---lt 4~

(I0 -- 4-y) 1---- L(( L L

~ ~ ~

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A -( LI a _ Iuuct + ~-(oo =-0

0

) (a- b ~ (q ~ LV) 0

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r

Of f

(

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4 r 17 l--~ r

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c

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~r ~~y v ~((II 11)11

~ k ) 1 ~ (JfJ )~ c

crf~J C-M--f1

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[t (v7 LOOaI l-5~~ )

Ivvu r Jl ik-- t I

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raquor- ~L 1

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(J r~ ~ ~ (~GII fP ( 1-5(~ j

h 60- ~~ Y 2-11gtV91

( ~ (~Jlt)~ ~ V(~ J ([a0

Y

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) ~ sect 51~(t7 7 LD lt) (t h~ 9- K -) 1ampt () gt plusmn fT-middot

Ji~ =- I J- ~l V t )

~ - AJD-

+ y2S0vpound 1

yL-shy~ J~sect3lk

LAv~ poundclt ~ 3- C6j 1--lt _ 1 fco+-~l ~~

gt 4- tofY IS fh l( lshyit

) iJ~L iA ~) 1~L( =j I

( (

)~ 2-lt 0 Y~r ~t r if I yen ) 0lt ~ zt-((1--- M~ ~((l--( ltt

~ - Al~

(

-) -fea)--z

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o~~~-k ~ fc wl v~f ~ c [Csez~ ~ (~tZ J- f r2

~) ~ gt~ Lv~ ~~ c lt2 0 -k ltlt - 1 r Y L~ 1

- I)( 1~ J plusmn- v o1-t-q gtL

1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

iJ-J~-TrrgtJA (~I~)

c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

) -- shy-~I~-

( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

(

l7 cent e l -

~)olt ) I_qshy

1e Jshy

lt (

) I0c shy 4shy 7 -) q L gt ( 4 ft 0)

o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

1- C(J~ ~ Y- flh~ ~ 1A- or )shy

UICvlgtlt-gt1tid (+(1-) ~

L~ 51 rf- A~~ f r 1

(2- i t- ( f ~ 7~ - i~

7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

-) p~ ift ~ It lt----- or ~ -r

-) 0deg1 f ~ AjL

(

_ ( L5f - ( 1

T~)L+ ~) L C

tIffiI

~ rmiddot- ~ ~ tf7~ ~ T

-) r laquo) y -r- tG2sf - ~ ~ o

-( ( verY ~ -- q 1 V 60 1G -t

q

I

( c1r J r-C A (

r - T I - v- 1

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~ ltL ~~ i~ r ~ t rlr~~ c1Vo ~ k~ SJd~u ~ 1a

~) 1 t~-t ~L f- J2Lf1--pound~tL z 2Q ~ q z I~

cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

pound ~ fJoQ ~~ (V CO IU( 2- f I iJ1-~ 1 AL cent - Y - I

i (~cent~ - y~ (u _LJ -1 ~~ Llt)9shy

~ V5 ~ (

~[-ch)( shy(~I +00 ~~

tL t __ ~ vr

7) -1 (20-- ~~ (1--6sLJ ) gt

p 4- c a Jb + iffLltgtrJ- l (r- C ) J~J- 0

) C -bU 0 f 7) 0 xG- 111L (

f 0

( o r

r 4-(0)19 -f- IIgt - )Le-f4 s ~

iJ LofO Ifplusmn~ 2J)s Jxi

f r- ~

)

-+~-~(-J

~( - 1 r (i(~o) J-shy L

~

I

etto I)

) L -+ y 1-_ I

(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

--1deg1 I feJ-q ~ to 1to (()) ~ljllLlvJ7htshy+ ~J Q lt= ~ ~ (fyengt-)

WmiddotI-c oc o - 1plusmn j t

- -

) r l

u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

- JJ

~I L

(j~l -t ~i~1 ( ) r-r --1 - ~U(

-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

-shy

-) C( ~~2i b ~ f--J

11 1Jamp 71 Ovb z J iT1 ~ 3

~~ FY 6 0Lu cJ t ~ 1-l-Y = b~

m~ v~ == ~

-- at ~ 1 +y~ co)

~-S-~_DT~O bull -~s-~+ff~

( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

~ l~ pnr -lNX Nw~middot oj 0 r~

f hiUUA 1N- d ~ p oJshy

~~ ~ Am Lb)

- --

~

~ ~

f ~ n

~o

re ~~ ~ I P tJU ~~ J Y --0

tf 1 ~ r t ~ ~

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J t ~

y 5-~ ~ ~ )J ~ 1

rgt l~ t--- ~ craquo ~ gtJ - 11 ~ ir shy

~ 11 ~1 jI

i ~

1 tv-- ~ t ~

J I - + 0 ~r +9

0 - 1 LYV t ~

IT ~~~ ~ CO ~~~

~ r-laquo r v1

1t jJ ~ )gt 11 t ~ g

J JJ ~

-- middot~1~ ~ ~

~ r (gt

(p -J

~ ~ ----

L--S- ltraquo~~ ~

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~-~

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l k1 ~ h elii~

~ (Jrc l1rrT ~ ~

lj 2shy3m + ~m ~l ~O

( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

~~ ~ c t-7(g t~ +PI P2) d-BMu

~ 1- L + ~) 1 (b )J- ~

z ~b ~ hrVgt (A)

~ YO f p~ ~ E

AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

U 2shy-z) ~- e f

~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

- --

N A-rR))( M A--TLJ--

~Wamp( b ~d

~ + ~~ -o-- pl-shy

cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

-gt ~ -z ~ -t S4 10 )- 0 l 10 gtshy

cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

LotLU ~ ~meh-~middot

w ~Y etmiddot ~ v

- -

~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

~ - l 0 ()-Z7- + ----) ~ --- + ~ ==-1 0)- 10)- ()L kshy

-) 0- ~ ~ r bi--- ~

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if9

(

0 I-e_

cJJishy middot 2- 102-)t e-~ -17] ~~

~)= ~-- 1 - e2- C c~

a)shy

e -- -) -2shy

( c) ~f2- k~ ~

D) ~ f ~ ZoJ-- a 2-b- Q

et--Q =- ~ ~ ~ ~ bullI

a2--e 1z (1 ~ ~ 2-- tL t - 2 C

(OJ I~rt--

-t

tf

~ ~--C

~~ t-

i ~ ~

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+

~ I

)+ ~

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~ poundshy g~ i ~

__~~

i ) ~~ ~ ~ ~ oot ushy

~ 5

l)h cstr bullbull~

(

1 ~ ~ Cpound Q) 1Jgtl + ~ ( tl i-~~-~middotf _~ (~_ M ~Q

1

- 1

rp ~

-shy y

~ 1 t l

Y C 1

Yr I I rl~ (

1 - -gtr IJ~~1 ~ tr 0 r i

~

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1

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~ middotu~

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camp (

1shy-4

r

bull

~-

(-

~ - - ~-- ~lt

~t+ ~-~ + ~ ~r) ~ -~

--(

~~ f ) - ~f-)--~~

~

-- ~amp~yen-- Jl

I

lti)shy

gt- to ~ -(

- Imiddot~cal --~~ ~

( 4 -- ~

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tpound- ~ ~ JLi~ Ji

f~d r L~ ~ flshy

~ S+ A ~ f~ _ r J~

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al~LL-=shy

c middot~ rc shy ~

(

~ -~Q

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(

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j~f ~ -shy ~ ~

s~-Q - 3 ~ )

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(

aL -

(

Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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(-04) d l bl-Ja ) ~

QIshy 01-shy ~~

l~ l~)

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A ~ ~3~ - ~

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ltxt~

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l shy

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-shy- 35

~(shyI lt0 ~ ~ ~ Q

+) ltS~ -- 51 ~gt Ztmiddot~-~~middotmiddotmiddotmiddotmiddotmiddotmiddot

- 2i 9A~

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J 4ltr~r ~b~~~~ ~

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middotle~ -lts --~bull shy

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i

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3

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~ r-_ (~-~)

~ NVD ~ ~~ Y-

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bo 0 - ~ - shy -) ~ amph~ - ) ~~ ~

_--_ _

r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

I~ -- ~ L - ~ 1shy

d ~ YrI1 ~ ~ ~ VY1 t- L

~ ~~ 1h6)+ C ~)

~ ( - roo rt )- - gtV1- 1- L

) -) ~l- _ V1 - r ~ tI 4- l-

~ k 9 ~ -egt~ laquo (f = lshy J~ -0

I ~

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R ~ n~)L ~

~ ~- -~-)

~~ ~v~-)

lC1-rtL l b)

- lP-~ ~ l~L - ~-) -~

yenl ~n L ~~~ ~( ~ ) ~ ~ K

t~

v o0

~

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t ~ ampb -t

~bull i -lib- ~

~ ~

r- ~

j

cp ~

~

~

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CD

( - YJ3 -~) [3 3gt

JC -shy t 5l f -1JI

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( gt

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-i-l~~J 2 0 46

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Q

(- ~

s

j O~ eqr~~tFl1~~~b __c~_~-

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-m ~_L tl- J

(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

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(

T

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lUI JI -f n~ (j

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JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

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o ( arltJ)

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~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

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( Il v ---t~ = ~ 01f)

f (-i~~)

ampSJ J--~~ ~~

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u )lt L L

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(I0 -- 4-y) 1---- L(( L L

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r

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h 60- ~~ Y 2-11gtV91

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Ji~ =- I J- ~l V t )

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) iJ~L iA ~) 1~L( =j I

( (

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~ - Al~

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~) ~ gt~ Lv~ ~~ c lt2 0 -k ltlt - 1 r Y L~ 1

- I)( 1~ J plusmn- v o1-t-q gtL

1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

iJ-J~-TrrgtJA (~I~)

c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

) -- shy-~I~-

( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

(

l7 cent e l -

~)olt ) I_qshy

1e Jshy

lt (

) I0c shy 4shy 7 -) q L gt ( 4 ft 0)

o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

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L~ 51 rf- A~~ f r 1

(2- i t- ( f ~ 7~ - i~

7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

-) p~ ift ~ It lt----- or ~ -r

-) 0deg1 f ~ AjL

(

_ ( L5f - ( 1

T~)L+ ~) L C

tIffiI

~ rmiddot- ~ ~ tf7~ ~ T

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q

I

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r - T I - v- 1

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cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

pound ~ fJoQ ~~ (V CO IU( 2- f I iJ1-~ 1 AL cent - Y - I

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) C -bU 0 f 7) 0 xG- 111L (

f 0

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r 4-(0)19 -f- IIgt - )Le-f4 s ~

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f r- ~

)

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~( - 1 r (i(~o) J-shy L

~

I

etto I)

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(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

--1deg1 I feJ-q ~ to 1to (()) ~ljllLlvJ7htshy+ ~J Q lt= ~ ~ (fyengt-)

WmiddotI-c oc o - 1plusmn j t

- -

) r l

u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

- JJ

~I L

(j~l -t ~i~1 ( ) r-r --1 - ~U(

-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

-shy

-) C( ~~2i b ~ f--J

11 1Jamp 71 Ovb z J iT1 ~ 3

~~ FY 6 0Lu cJ t ~ 1-l-Y = b~

m~ v~ == ~

-- at ~ 1 +y~ co)

~-S-~_DT~O bull -~s-~+ff~

( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

~ l~ pnr -lNX Nw~middot oj 0 r~

f hiUUA 1N- d ~ p oJshy

~~ ~ Am Lb)

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~

~ ~

f ~ n

~o

re ~~ ~ I P tJU ~~ J Y --0

tf 1 ~ r t ~ ~

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J t ~

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rgt l~ t--- ~ craquo ~ gtJ - 11 ~ ir shy

~ 11 ~1 jI

i ~

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J I - + 0 ~r +9

0 - 1 LYV t ~

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1t jJ ~ )gt 11 t ~ g

J JJ ~

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(p -J

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L--S- ltraquo~~ ~

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~-~

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l k1 ~ h elii~

~ (Jrc l1rrT ~ ~

lj 2shy3m + ~m ~l ~O

( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

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~ 1- L + ~) 1 (b )J- ~

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~ YO f p~ ~ E

AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

U 2shy-z) ~- e f

~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

- --

N A-rR))( M A--TLJ--

~Wamp( b ~d

~ + ~~ -o-- pl-shy

cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

-gt ~ -z ~ -t S4 10 )- 0 l 10 gtshy

cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

LotLU ~ ~meh-~middot

w ~Y etmiddot ~ v

- -

~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

~ - l 0 ()-Z7- + ----) ~ --- + ~ ==-1 0)- 10)- ()L kshy

-) 0- ~ ~ r bi--- ~

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if9

(

0 I-e_

cJJishy middot 2- 102-)t e-~ -17] ~~

~)= ~-- 1 - e2- C c~

a)shy

e -- -) -2shy

( c) ~f2- k~ ~

D) ~ f ~ ZoJ-- a 2-b- Q

et--Q =- ~ ~ ~ ~ bullI

a2--e 1z (1 ~ ~ 2-- tL t - 2 C

(OJ I~rt--

-t

tf

~ ~--C

~~ t-

i ~ ~

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+

~ I

)+ ~

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~ poundshy g~ i ~

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i ) ~~ ~ ~ ~ oot ushy

~ 5

l)h cstr bullbull~

(

1 ~ ~ Cpound Q) 1Jgtl + ~ ( tl i-~~-~middotf _~ (~_ M ~Q

1

- 1

rp ~

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~ 1 t l

Y C 1

Yr I I rl~ (

1 - -gtr IJ~~1 ~ tr 0 r i

~

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camp (

1shy-4

r

bull

~-

(-

~ - - ~-- ~lt

~t+ ~-~ + ~ ~r) ~ -~

--(

~~ f ) - ~f-)--~~

~

-- ~amp~yen-- Jl

I

lti)shy

gt- to ~ -(

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( 4 -- ~

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tpound- ~ ~ JLi~ Ji

f~d r L~ ~ flshy

~ S+ A ~ f~ _ r J~

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al~LL-=shy

c middot~ rc shy ~

(

~ -~Q

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(

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j~f ~ -shy ~ ~

s~-Q - 3 ~ )

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(

aL -

(

Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

o Ioshy

(-04) d l bl-Ja ) ~

QIshy 01-shy ~~

l~ l~)

~ 1

A ~ ~3~ - ~

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~ 2q ~ ltrshyI

ltxt~

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(

l shy

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laquo~

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ltt-- ~ t~~~ ~~

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-shy- 35

~(shyI lt0 ~ ~ ~ Q

+) ltS~ -- 51 ~gt Ztmiddot~-~~middotmiddotmiddotmiddotmiddotmiddotmiddot

- 2i 9A~

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( ffshy lt bull

J 4ltr~r ~b~~~~ ~

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middotle~ -lts --~bull shy

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)~

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Qshyl- bL shy~ ( 1gt- shy

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i

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3

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_--_ _

r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

I~ -- ~ L - ~ 1shy

d ~ YrI1 ~ ~ ~ VY1 t- L

~ ~~ 1h6)+ C ~)

~ ( - roo rt )- - gtV1- 1- L

) -) ~l- _ V1 - r ~ tI 4- l-

~ k 9 ~ -egt~ laquo (f = lshy J~ -0

I ~

~ ~k ~~

R ~ n~)L ~

~ ~- -~-)

~~ ~v~-)

lC1-rtL l b)

- lP-~ ~ l~L - ~-) -~

yenl ~n L ~~~ ~( ~ ) ~ ~ K

( - YJ3 -~) [3 3gt

JC -shy t 5l f -1JI

Ql) (6 C- -- bS3 f3gt

( gt

( ~

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-i-l~~J 2 0 46

~b~ _ 0 -- b lt

Q

(- ~

s

j O~ eqr~~tFl1~~~b __c~_~-

~- ---gt B~~IA) ( ctOJo)gt -

-m ~_L tl- J

(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

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lshy

1 IttP

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I~II~

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(

T

7) d j ) + 1-1 L

2shy

(

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( ~~ 01shy ~~~

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lUI JI -f n~ (j

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JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

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o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

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~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

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( Il v ---t~ = ~ 01f)

f (-i~~)

ampSJ J--~~ ~~

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u )lt L L

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e) (~) ~ - c= i(yen~ ) _ (

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(

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(I0 -- 4-y) 1---- L(( L L

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r

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h 60- ~~ Y 2-11gtV91

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Ji~ =- I J- ~l V t )

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) iJ~L iA ~) 1~L( =j I

( (

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~ - Al~

(

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~) ~ gt~ Lv~ ~~ c lt2 0 -k ltlt - 1 r Y L~ 1

- I)( 1~ J plusmn- v o1-t-q gtL

1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

iJ-J~-TrrgtJA (~I~)

c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

) -- shy-~I~-

( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

(

l7 cent e l -

~)olt ) I_qshy

1e Jshy

lt (

) I0c shy 4shy 7 -) q L gt ( 4 ft 0)

o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

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L~ 51 rf- A~~ f r 1

(2- i t- ( f ~ 7~ - i~

7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

-) p~ ift ~ It lt----- or ~ -r

-) 0deg1 f ~ AjL

(

_ ( L5f - ( 1

T~)L+ ~) L C

tIffiI

~ rmiddot- ~ ~ tf7~ ~ T

-) r laquo) y -r- tG2sf - ~ ~ o

-( ( verY ~ -- q 1 V 60 1G -t

q

I

( c1r J r-C A (

r - T I - v- 1

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cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

pound ~ fJoQ ~~ (V CO IU( 2- f I iJ1-~ 1 AL cent - Y - I

i (~cent~ - y~ (u _LJ -1 ~~ Llt)9shy

~ V5 ~ (

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p 4- c a Jb + iffLltgtrJ- l (r- C ) J~J- 0

) C -bU 0 f 7) 0 xG- 111L (

f 0

( o r

r 4-(0)19 -f- IIgt - )Le-f4 s ~

iJ LofO Ifplusmn~ 2J)s Jxi

f r- ~

)

-+~-~(-J

~( - 1 r (i(~o) J-shy L

~

I

etto I)

) L -+ y 1-_ I

(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

--1deg1 I feJ-q ~ to 1to (()) ~ljllLlvJ7htshy+ ~J Q lt= ~ ~ (fyengt-)

WmiddotI-c oc o - 1plusmn j t

- -

) r l

u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

- JJ

~I L

(j~l -t ~i~1 ( ) r-r --1 - ~U(

-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

-shy

-) C( ~~2i b ~ f--J

11 1Jamp 71 Ovb z J iT1 ~ 3

~~ FY 6 0Lu cJ t ~ 1-l-Y = b~

m~ v~ == ~

-- at ~ 1 +y~ co)

~-S-~_DT~O bull -~s-~+ff~

( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

~ l~ pnr -lNX Nw~middot oj 0 r~

f hiUUA 1N- d ~ p oJshy

~~ ~ Am Lb)

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~

~ ~

f ~ n

~o

re ~~ ~ I P tJU ~~ J Y --0

tf 1 ~ r t ~ ~

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J t ~

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rgt l~ t--- ~ craquo ~ gtJ - 11 ~ ir shy

~ 11 ~1 jI

i ~

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J I - + 0 ~r +9

0 - 1 LYV t ~

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1t jJ ~ )gt 11 t ~ g

J JJ ~

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(p -J

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~-~

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l k1 ~ h elii~

~ (Jrc l1rrT ~ ~

lj 2shy3m + ~m ~l ~O

( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

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~ 1- L + ~) 1 (b )J- ~

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~ YO f p~ ~ E

AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

U 2shy-z) ~- e f

~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

- --

N A-rR))( M A--TLJ--

~Wamp( b ~d

~ + ~~ -o-- pl-shy

cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

-gt ~ -z ~ -t S4 10 )- 0 l 10 gtshy

cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

LotLU ~ ~meh-~middot

w ~Y etmiddot ~ v

- -

~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

~ - l 0 ()-Z7- + ----) ~ --- + ~ ==-1 0)- 10)- ()L kshy

-) 0- ~ ~ r bi--- ~

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if9

(

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cJJishy middot 2- 102-)t e-~ -17] ~~

~)= ~-- 1 - e2- C c~

a)shy

e -- -) -2shy

( c) ~f2- k~ ~

D) ~ f ~ ZoJ-- a 2-b- Q

et--Q =- ~ ~ ~ ~ bullI

a2--e 1z (1 ~ ~ 2-- tL t - 2 C

(OJ I~rt--

-t

tf

~ ~--C

~~ t-

i ~ ~

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+

~ I

)+ ~

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~ poundshy g~ i ~

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i ) ~~ ~ ~ ~ oot ushy

~ 5

l)h cstr bullbull~

(

1 ~ ~ Cpound Q) 1Jgtl + ~ ( tl i-~~-~middotf _~ (~_ M ~Q

1

- 1

rp ~

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~ 1 t l

Y C 1

Yr I I rl~ (

1 - -gtr IJ~~1 ~ tr 0 r i

~

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camp (

1shy-4

r

bull

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(-

~ - - ~-- ~lt

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--(

~~ f ) - ~f-)--~~

~

-- ~amp~yen-- Jl

I

lti)shy

gt- to ~ -(

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( 4 -- ~

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f~d r L~ ~ flshy

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al~LL-=shy

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(

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(

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(

aL -

(

Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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QIshy 01-shy ~~

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A ~ ~3~ - ~

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ltxt~

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l shy

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-shy- 35

~(shyI lt0 ~ ~ ~ Q

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lC1-rtL l b)

- lP-~ ~ l~L - ~-) -~

yenl ~n L ~~~ ~( ~ ) ~ ~ K

~b~ _ 0 -- b lt

Q

(- ~

s

j O~ eqr~~tFl1~~~b __c~_~-

~- ---gt B~~IA) ( ctOJo)gt -

-m ~_L tl- J

(Uh)

~ AB ~~

900 ot ~

---- - ~----------~----- --------------- -------~=---

_11 bullbull ~ ~ ~ ~ J __ ~

r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

~~ of ~(Mrt ~ tl~ F~ bu~ t

~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

_

lshy

1 IttP

_~ 1 10-lJV

( C0)9shy~

JJ1shy-- - I+2shy I

I~II~

- =ULshy

(

T

7) d j ) + 1-1 L

2shy

(

Mshy

-11 J~t~) ~

( ~~ 01shy ~~~

0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

~ Vt~ -Lp o~ ~+flt Ik-- ~11c

~c f fGfshy

~~(+ 1A- ~ru f~ fCed t Y~-J(R-o) ~o

IL fnJ u~ lk~ ~~~~~ ~cf(l- ~ lt) C~r~ cf ~-k~J (

~~ Y 0+-) shy~ J rO)tJ~ BY)-~e- poundLlr aP---- ~

f ( O-Lc-W I ~~tL9) (( 0 c )~ f ~ J (gt -it (oreUJ I

o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

e- I0 6 [)~i(P-UJ Cort - Cr(amp -fJ -1~ o]

~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

I yr~) Ci( (~ z I

ViJ-e plusmn = a

r(~ CR~c-r f~~ (0 (C) 0 r (0 (-oj

~ -- ArC

( Il v ---t~ = ~ 01f)

f (-i~~)

ampSJ J--~~ ~~

-(9 (

~vM J (+go) e (~-1(o)

~~ J ~ -fJrvtf

u )lt L L

T~+ =zt

e) (~) ~ - c= i(yen~ ) _ (

J( z cit fLe- rLf~ ~( ~ e ~

tgt poundL- it

~ - 4-- +c9 - + rL -7 ~ c~ - cshy

0) Co-ho-= i -( s z- h=- plusmn+(~

Al~ 19 ~ $ +lt ~~ = -f if t I

(

)IL Q

-i+f-~1

]~ QV)~ ftt u~ ( q-_ r) ~

Ji-fl~f Cba b)

(

~ )Itl t L - 1--_ - by + YL--~

aLshy

1~J A ~ o __4y 1 kl---lt 4~

(I0 -- 4-y) 1---- L(( L L

~ ~ ~

o-shy

qD 2 _ -+~ 1-( (cJJ _ G) = o

A -( LI a _ Iuuct + ~-(oo =-0

0

) (a- b ~ (q ~ LV) 0

-) Q1_ l -ashy 1Jo f Q -- 2 cJ

r

Of f

(

t k yfl-=zl 1

eJlJ ~trJ J ) t r-~ ~ y ~ ~r Jro-alt-I bshy

j I euroj-J~j ~ Y~ -4JL rTr1+ 2 -4x t ~

~ J I Y-+ 4gt i+--v

+N - Hshy

4 r 17 l--~ r

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tl

lwt ~ 0 ~J11 yo -~ ]I j Jaf)l~)+ Oit) ~---~~--

~i~ ~ V~ ~(~+l J 4 J 1- =I (~( ~ ) t

c

0 -t it ~ z -+- gt~

(4- x~)lt CJ ~J ~ I

S T ~ (+-1 I1 )

ampf~~ -JrJ~ 7lt0 - gt j ~O-yl T ~)

~ - ~ ~ (11) (

~gt C4y~~~) ~ f S-

~r ~~y v ~((II 11)11

~ k ) 1 ~ (JfJ )~ c

crf~J C-M--f1

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Yr =- plusmn (YI( 1

- l L

T~ f ~r

-p-t ( p-lshybe- (0 Ca)Xgt tM~

[t (v7 LOOaI l-5~~ )

Ivvu r Jl ik-- t I

(

raquor- ~L 1

~ qr~ t

(J r~ ~ ~ (~GII fP ( 1-5(~ j

h 60- ~~ Y 2-11gtV91

( ~ (~Jlt)~ ~ V(~ J ([a0

Y

shy

) ~ sect 51~(t7 7 LD lt) (t h~ 9- K -) 1ampt () gt plusmn fT-middot

Ji~ =- I J- ~l V t )

~ - AJD-

+ y2S0vpound 1

yL-shy~ J~sect3lk

LAv~ poundclt ~ 3- C6j 1--lt _ 1 fco+-~l ~~

gt 4- tofY IS fh l( lshyit

) iJ~L iA ~) 1~L( =j I

( (

)~ 2-lt 0 Y~r ~t r if I yen ) 0lt ~ zt-((1--- M~ ~((l--( ltt

~ - Al~

(

-) -fea)--z

~kJ -7 (t lI ~ -= sectr~(~ rNJ)cgf-e ~ 1 SU-lt

o~~~-k ~ fc wl v~f ~ c [Csez~ ~ (~tZ J- f r2

~) ~ gt~ Lv~ ~~ c lt2 0 -k ltlt - 1 r Y L~ 1

- I)( 1~ J plusmn- v o1-t-q gtL

1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

iJ-J~-TrrgtJA (~I~)

c----shy~ I - I ~ -1 +- 1 Q 1-( 0 S~

) -- shy-~I~-

( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

(

l7 cent e l -

~)olt ) I_qshy

1e Jshy

lt (

) I0c shy 4shy 7 -) q L gt ( 4 ft 0)

o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

1- C(J~ ~ Y- flh~ ~ 1A- or )shy

UICvlgtlt-gt1tid (+(1-) ~

L~ 51 rf- A~~ f r 1

(2- i t- ( f ~ 7~ - i~

7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

-) p~ ift ~ It lt----- or ~ -r

-) 0deg1 f ~ AjL

(

_ ( L5f - ( 1

T~)L+ ~) L C

tIffiI

~ rmiddot- ~ ~ tf7~ ~ T

-) r laquo) y -r- tG2sf - ~ ~ o

-( ( verY ~ -- q 1 V 60 1G -t

q

I

( c1r J r-C A (

r - T I - v- 1

~ -- (ArD)

~ ltL ~~ i~ r ~ t rlr~~ c1Vo ~ k~ SJd~u ~ 1a

~) 1 t~-t ~L f- J2Lf1--pound~tL z 2Q ~ q z I~

cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

pound ~ fJoQ ~~ (V CO IU( 2- f I iJ1-~ 1 AL cent - Y - I

i (~cent~ - y~ (u _LJ -1 ~~ Llt)9shy

~ V5 ~ (

~[-ch)( shy(~I +00 ~~

tL t __ ~ vr

7) -1 (20-- ~~ (1--6sLJ ) gt

p 4- c a Jb + iffLltgtrJ- l (r- C ) J~J- 0

) C -bU 0 f 7) 0 xG- 111L (

f 0

( o r

r 4-(0)19 -f- IIgt - )Le-f4 s ~

iJ LofO Ifplusmn~ 2J)s Jxi

f r- ~

)

-+~-~(-J

~( - 1 r (i(~o) J-shy L

~

I

etto I)

) L -+ y 1-_ I

(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

--1deg1 I feJ-q ~ to 1to (()) ~ljllLlvJ7htshy+ ~J Q lt= ~ ~ (fyengt-)

WmiddotI-c oc o - 1plusmn j t

- -

) r l

u(

C~-~) - 4- V 2 [j-- C- --U) -) X

L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

- JJ

~I L

(j~l -t ~i~1 ( ) r-r --1 - ~U(

-) raquoL -] n( =- fJ -+ n -

(

(

~ Gt 1J~ ps l5 Rs = (

(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

-shy

-) C( ~~2i b ~ f--J

11 1Jamp 71 Ovb z J iT1 ~ 3

~~ FY 6 0Lu cJ t ~ 1-l-Y = b~

m~ v~ == ~

-- at ~ 1 +y~ co)

~-S-~_DT~O bull -~s-~+ff~

( r ldJl-P) ( ~~-Y

tgt)- shy ~

Mo dMkvu bjw r J Afi~~ ~

(A)-- b~ v L~~ b~r- )

a-~ q 2 2( mf~ IJl ~+~~l

~ l~ pnr -lNX Nw~middot oj 0 r~

f hiUUA 1N- d ~ p oJshy

~~ ~ Am Lb)

- --

~

~ ~

f ~ n

~o

re ~~ ~ I P tJU ~~ J Y --0

tf 1 ~ r t ~ ~

~ rgt~f 11

J t ~

y 5-~ ~ ~ )J ~ 1

rgt l~ t--- ~ craquo ~ gtJ - 11 ~ ir shy

~ 11 ~1 jI

i ~

1 tv-- ~ t ~

J I - + 0 ~r +9

0 - 1 LYV t ~

IT ~~~ ~ CO ~~~

~ r-laquo r v1

1t jJ ~ )gt 11 t ~ g

J JJ ~

-- middot~1~ ~ ~

~ r (gt

(p -J

~ ~ ----

L--S- ltraquo~~ ~

~~ t

~-~

~~ ~ 1- f-h -~~-------I~-r---

l k1 ~ h elii~

~ (Jrc l1rrT ~ ~

lj 2shy3m + ~m ~l ~O

( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

- lt

l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

~ ~ )(4middot+2 1 f _ 1 Aw (A) I-1 r~ -- --c- 4shy1( lu (2 L L ~ rin

middot--_-_~-- -_-----_ ------- __----- --shy

(J2 A)Y-- 1 amp veo-oJJ ~cJ PIE I Egt- P Q)

i2A PI E I Egt- P M ~r- u-M

~~ ~ c t-7(g t~ +PI P2) d-BMu

~ 1- L + ~) 1 (b )J- ~

z ~b ~ hrVgt (A)

~ YO f p~ ~ E

AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

U 2shy-z) ~- e f

~ Le- egt JJ--z (

37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

- --

N A-rR))( M A--TLJ--

~Wamp( b ~d

~ + ~~ -o-- pl-shy

cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

-gt ~ -z ~ -t S4 10 )- 0 l 10 gtshy

cry )(2- + f _ 1- o ~ b- b

(

~= a~CI-e) ex

m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

0- 2-- c

LDtuJ in f~~

Lou-u M M-I ~1 QlltLshy

LotLU ~ ~meh-~middot

w ~Y etmiddot ~ v

- -

~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

~ - l 0 ()-Z7- + ----) ~ --- + ~ ==-1 0)- 10)- ()L kshy

-) 0- ~ ~ r bi--- ~

__ ~ S--

if9

(

0 I-e_

cJJishy middot 2- 102-)t e-~ -17] ~~

~)= ~-- 1 - e2- C c~

a)shy

e -- -) -2shy

( c) ~f2- k~ ~

D) ~ f ~ ZoJ-- a 2-b- Q

et--Q =- ~ ~ ~ ~ bullI

a2--e 1z (1 ~ ~ 2-- tL t - 2 C

(OJ I~rt--

-t

tf

~ ~--C

~~ t-

i ~ ~

~ ~ r

+

~ I

)+ ~

~

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~1-

~ poundshy g~ i ~

__~~

i ) ~~ ~ ~ ~ oot ushy

~ 5

l)h cstr bullbull~

(

1 ~ ~ Cpound Q) 1Jgtl + ~ ( tl i-~~-~middotf _~ (~_ M ~Q

1

- 1

rp ~

-shy y

~ 1 t l

Y C 1

Yr I I rl~ (

1 - -gtr IJ~~1 ~ tr 0 r i

~

QI~ t

1

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shy

~ middotu~

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)~L c l~ kgt) - --

camp (

1shy-4

r

bull

~-

(-

~ - - ~-- ~lt

~t+ ~-~ + ~ ~r) ~ -~

--(

~~ f ) - ~f-)--~~

~

-- ~amp~yen-- Jl

I

lti)shy

gt- to ~ -(

- Imiddot~cal --~~ ~

( 4 -- ~

b ~ VS -

tpound- ~ ~ JLi~ Ji

f~d r L~ ~ flshy

~ S+ A ~ f~ _ r J~

~ 1shy--

al~LL-=shy

c middot~ rc shy ~

(

~ -~Q

~VP~

(

~ _c~ C ~jlt1__ ~U-

j~f ~ -shy ~ ~

s~-Q - 3 ~ )

~ I

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(~j~~ ~- -r- shyCd ~-J

(

aL -

(

Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

o Ioshy

(-04) d l bl-Ja ) ~

QIshy 01-shy ~~

l~ l~)

~ 1

A ~ ~3~ - ~

~ ( ~f~t

~ 2q ~ ltrshyI

ltxt~

~ -shy

~ ~ F

(

l shy

- ~- J~~ - ~ 13~- --~~

lt

~I ~ ~ ~ ~ ~ 1-~-J~ 1shy - h- i

3

laquo~

~ raquo ~

gt

(

(j gt~ - - ()

~L ~ ~ ~

d- ~)-

1t

~

e - L1-

N ltk~f- ~

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-- - - - -

- -- -shy

~ )

Is ~)-t)~~~ ~~

0shy

--~- - --_ _ --

~ ~L (f-L ~t- ----shyl-~ Iamp

~~ ~~) ~ + ~ -

ltt-- ~ t~~~ ~~

~~ 1 ~~ -- 1shy s

-shy- 35

~(shyI lt0 ~ ~ ~ Q

+) ltS~ -- 51 ~gt Ztmiddot~-~~middotmiddotmiddotmiddotmiddotmiddotmiddot

- 2i 9A~

~ S ~

( ffshy lt bull

J 4ltr~r ~b~~~~ ~

0 j~ l~S ~~ ~~~~)shy

middotle~ -lts --~bull shy

-= - h ttll ~)

)~

0) ~ -shyb)

-shy~)- ~

L -

Qshyl- bL shy~ ( 1gt- shy

~ ~ - L

) ~ -~-

~~---l lt bull lmiddot~ u

)1--bull~~ ~~~bull( -]3 shy

-X(lt--

i

-

~~~l~ ~~~ft

--=--~==========-ft1rE3~ ~=-

j-4-~ ~tIl-~

~=~

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~ ~~ ~ ~~J ~

)b

)

4shy

-t C g

c ~ ~~OuI

(

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3 C) - )shy +shy ~ C~- t-)shy ~ J-~

( R~~L ~ l-)l -

5 9

reg

~ A T C -t ~ - ~ 0 - s ~~

( YoT J--~ - ) i- ~ ( d - IJo l-) ~- l r~

r--shy

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L~

l shy -

(

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t J(~~lL +lt1 - ~-_

~

(

J -- -- ~- Jb 2gt shy

~ ~

regshy

1~ (~ --Jshy + Ltj (a-~)L ~

~ t

l shy L))shy shy (rl _ ~)L shy

~

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J- 0

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1

(

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-C 1-~)

(~~~ +~ - ~l-_ ) ~

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~ ~1gt ~~~ ~~ ~

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~ Cb(~)

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4- ~Y ~V ~rlt ~l- -~

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middot-_ _ --shy

c $-b

i

( shy

Q- Cl ~ ) ~S~

( ) ~ l~t~~) ~~Q

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~-_ f- bullt C)( ~-

5~f~ -shyuJ)v ~

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_ (

(

-rr4

(

~ ~

q~

_~---__-__-----~~_ - _ _- _ _--_ _

r-_ ~1

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( c ))

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---

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t laquo rI ~u JJ f I~

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t ~u~middotmiddotmiddotmiddotcJ ~~~ ~

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GD- ~~~ ~~

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~shyQfshy

t( i ~ - (~ li)0

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t

oL ~-- ~ -L o )7- C) l tgt

rmiddotmiddot

raquo

a i - 0- e

l OL 0~ 01 - o-Q 1 -= shy

h ~_ ~ ~e~~~ ~

( ~ - J

~ 7 ~ ) h ~1~ tl- v -

-~ ~~~~~- ~ ~ th I d- ~ Jgt~ ~~

~

__----==- _ __ o-c=oco~--middot= = _ -__ _ __ ___~ ==--=__== Jo1 rfI~t ~

~~ et~

- _4- ) + 3L

3

- 4- ~ -t-~-~

3 ( -t- 4- 1 ~ )-1

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~-- )-~ ~~~i

~ aL- ~ Qk

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( G ~ ~

CC~ - ~)- ~~ ~(1( ~ ~ - L

) 06 ~~-~~ )

- l~~)

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(~~-) ~

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1 (L-_~) 1- l(tli- ~-) - L

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middot 1 L bull1 h~

(

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~ r-_ (~-~)

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bo 0 - ~ - shy -) ~ amph~ - ) ~~ ~

_--_ _

r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

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d ~ YrI1 ~ ~ ~ VY1 t- L

~ ~~ 1h6)+ C ~)

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) -) ~l- _ V1 - r ~ tI 4- l-

~ k 9 ~ -egt~ laquo (f = lshy J~ -0

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~ ~k ~~

R ~ n~)L ~

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~~ ~v~-)

lC1-rtL l b)

- lP-~ ~ l~L - ~-) -~

yenl ~n L ~~~ ~( ~ ) ~ ~ K

---- - ~----------~----- --------------- -------~=---

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r

~ ~t~ =- 9M1 ~ L Cshy

bull2- Lshy

(

bull bull

I ~

~

~ -(L

n o

~rlt ~ ~ ~ i7t e~~-

-

+ +

p

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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c~r~2 Pr c-reg

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o ( arltJ)

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( Il v ---t~ = ~ 01f)

f (-i~~)

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u )lt L L

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1+4u~~ 4 ~ k d vo~--Z + laquoe-~ uA~

iJ-J~-TrrgtJA (~I~)

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) -- shy-~I~-

( r ) 4- ~ C( ~ 1bL

QL-t cd) 1--= 4- ~

1Jl AM~111

oL-f-Cj bL 306~

)

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l7 cent e l -

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1e Jshy

lt (

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o ~ t ~)u( Jyen-t ltgtltgtJ 1c ~ k-- 7 coL L 0 --) rraquo L ~ a

ff+ (~) ~4c ~jllcl)

fJ - 0 ~rltf ~ zdL)~~

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7 Si~Q ~ 111) 2 i 1)

(-- shy(2shy

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(

_ ( L5f - ( 1

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I

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cn-J lCltk- ~ Ie t ~ L t fJ1-1l-shy

(

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f r- ~

)

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etto I)

) L -+ y 1-_ I

(71 0 ~ (- Vl - n () ~J =- (+ J3 - -~ )

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WmiddotI-c oc o - 1plusmn j t

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) r l

u(

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L

+ L1ll1c 0[1- 0 ] ~ ~ J1 (I

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(

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(I~ ~i ~f~~ ~ ~~ ~ middot~t

~~~~=

C ~~ f A l L f 1 D llhe 0]- I J1 e gt~

jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

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m~ v~ == ~

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( r ldJl-P) ( ~~-Y

tgt)- shy ~

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( rn~- ~4-) rr ~ t )~

S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

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l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

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middot--_-_~-- -_-----_ ------- __----- --shy

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AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

-~ ~ h o1tv o r Mffz-ti ~rJ ()

~d~D ~ l0

9 n ~ tJwmJJJ ~ (~f ~)

Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

N~(Y1~ f~ ~ (6-~)

GLb (jc---eJ

(JY e 2- b w oshy

9 elJ~ l-e2shy

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37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

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N A-rR))( M A--TLJ--

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cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

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cry )(2- + f _ 1- o ~ b- b

(

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m K _ lJ1shy- Gl a 1shy

1 ( -==1 l - - 0

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w ~Y etmiddot ~ v

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~

CAe) p~~ 10) e (o-f)) )Ai on ~f~

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l)h cstr bullbull~

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1

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rp ~

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r

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aL -

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Ja~- ~ kst -~

50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

o Ioshy

(-04) d l bl-Ja ) ~

QIshy 01-shy ~~

l~ l~)

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A ~ ~3~ - ~

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ltxt~

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l shy

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-shy- 35

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- 2i 9A~

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middotle~ -lts --~bull shy

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p

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22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

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c~r~2 Pr c-reg

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~~~~=

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jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

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S LlM latA) 12- ( 4 ~ Lr) ~ E ~ (-2 -I)

poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

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LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

C)f corraquo F~~M ~ UA ( = Y

Ne ~t (-40)

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middot--_-_~-- -_-----_ ------- __----- --shy

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AS$~RTION ANDRIASON~

( ~-

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J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

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(

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m K _ lJ1shy- Gl a 1shy

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aL -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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ltt-- ~ t~~~ ~~

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-shy- 35

~(shyI lt0 ~ ~ ~ Q

+) ltS~ -- 51 ~gt Ztmiddot~-~~middotmiddotmiddotmiddotmiddotmiddotmiddot

- 2i 9A~

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J 4ltr~r ~b~~~~ ~

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-shy~)- ~

L -

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i

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reg

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t J(~~lL +lt1 - ~-_

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i

( shy

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( ) ~ l~t~~) ~~Q

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t ~u~middotmiddotmiddotmiddotcJ ~~~ ~

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~shyQfshy

t( i ~ - (~ li)0

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t

oL ~-- ~ -L o )7- C) l tgt

rmiddotmiddot

raquo

a i - 0- e

l OL 0~ 01 - o-Q 1 -= shy

h ~_ ~ ~e~~~ ~

( ~ - J

~ 7 ~ ) h ~1~ tl- v -

-~ ~~~~~- ~ ~ th I d- ~ Jgt~ ~~

~

__----==- _ __ o-c=oco~--middot= = _ -__ _ __ ___~ ==--=__== Jo1 rfI~t ~

~~ et~

- _4- ) + 3L

3

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3 ( -t- 4- 1 ~ )-1

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) 06 ~~-~~ )

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bo 0 - ~ - shy -) ~ amph~ - ) ~~ ~

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r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

I~ -- ~ L - ~ 1shy

d ~ YrI1 ~ ~ ~ VY1 t- L

~ ~~ 1h6)+ C ~)

~ ( - roo rt )- - gtV1- 1- L

) -) ~l- _ V1 - r ~ tI 4- l-

~ k 9 ~ -egt~ laquo (f = lshy J~ -0

I ~

~ ~k ~~

R ~ n~)L ~

~ ~- -~-)

~~ ~v~-)

lC1-rtL l b)

- lP-~ ~ l~L - ~-) -~

yenl ~n L ~~~ ~( ~ ) ~ ~ K

2 2

22 Eqaatien of ellipse is ~ + YI =1 penetal t~ent to the ~1IP$e of sl~pe m is y = mx +

J2ml +1 ~ Since thisis~tl~liy inclined to axes so m = +ti~1PllsO~gel1ts are

y=+ x+J2+1=+x+Jj

nfMo~of any oftKesll tabgent thesefr6morigiifigequaltltl~ ( lt lt ~~~ii

Hert1ce~a)lstheC(frte6t~vVet

gt d bull bull ii 1 _ - ltiJ~ ~ 34 Eluationt(ftangentls~ ~+L_= landequati(1)noftaagentattne~1Jomteuroacos+bsln~ ) IS a2 b 2 i - - )lti bull

gtit~tf~smjl=I

4-1 ~ ~ ~ 1Bothare-seme le cosegt 2 Slnr = 2

1t

= lt1gt=6

Hence (aisthe correct-answer

t~

~r l

bull --cl )

LY == A-X _ ~ It)

~~ of ~(Mrt ~ tl~ F~ bu~ t

~- o~ ftJshy ~

c~r~2 Pr c-reg

~ e ~ (Gt1lt U v ~ O)-rI-shy D

J- -shyr gt

_

lshy

1 IttP

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( C0)9shy~

JJ1shy-- - I+2shy I

I~II~

- =ULshy

(

T

7) d j ) + 1-1 L

2shy

(

Mshy

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( ~~ 01shy ~~~

0 b I il0f-n-rvvr ~rC) ~ (+_ OJlt- ( -tshy 602- )

lUI JI -f n~ (j

-=) ~L = a2- -+ oj

i t a ( c 1j ~- ~~rg

JltL L -l-(1gt J middotf

bull I (P ) (tv) Qv1 ~-e- poundlU(gtsc- 1 0 I - -( la= 2ltamp-=-- 12shy~ f-- J 4middot~LC4

I ~ f- -6 0 1 r

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~c f fGfshy

~~(+ 1A- ~ru f~ fCed t Y~-J(R-o) ~o

IL fnJ u~ lk~ ~~~~~ ~cf(l- ~ lt) C~r~ cf ~-k~J (

~~ Y 0+-) shy~ J rO)tJ~ BY)-~e- poundLlr aP---- ~

f ( O-Lc-W I ~~tL9) (( 0 c )~ f ~ J (gt -it (oreUJ I

o ( arltJ)

degremiddotLl pOeJogt j ( OCilJ1-[pound~(D-o()J-o cT()-f-ltjJ1Iltoj

e- I0 6 [)~i(P-UJ Cort - Cr(amp -fJ -1~ o]

~ 1 ~ i~ (6)+- ampJ I ~ 9 fJ AtA a

--

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

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_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

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_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

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=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

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a6h 2 - a2b

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(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

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( _1 +_1_+_1 +_1__)= EXtX2X 3

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)

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(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

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_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

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()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

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)

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(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

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middot--_-_~-- -_-----_ ------- __----- --shy

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AS$~RTION ANDRIASON~

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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~~~~=

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jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

NIW S4 1 ~~ ~ OM- 0 C R~) (A- Q

( f ~l DQ~) E- ([)D l S p) ~ CA1~Llb) ~ ~

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-

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LU- N M (h1lt)

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olaquo k d-2- gt( - 2-4

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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~~~~=

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jb U~L ~ t Q M J- ~

ecc~C- ~ i Pc s e P )71-0 1tt-Beuro 9Jf1~e-

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poundg PI r- - z Jb~ -[4+0 J- = 161 (

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l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

4gt ~middotk ~ 2 (x + k)

olaquo k d-2- gt( - 2-4

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Ne ~t (-40)

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middot--_-_~-- -_-----_ ------- __----- --shy

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AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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poundg PI r- - z Jb~ -[4+0 J- = 161 (

-

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l4 r ~ ~d P fgt2 ~ ~ J

LU- N M (h1lt)

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olaquo k d-2- gt( - 2-4

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Ne ~t (-40)

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AS$~RTION ANDRIASON~

( ~-

-- JI 2 3 - tb - - Jt t 1 J- 1middot

J bull~ --J X ~ - 7 ~ ------~ ))Vm m ~ j~ -1)

(L - -

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~d~D ~ l0

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Oe~ ~ _ ~ = 02- b) (jJL VjlAshy

a _a dJO a-eshy

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(JY e 2- b w oshy

9 elJ~ l-e2shy

U 2shy-z) ~- e f

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37 ANgt (C-)--- o huJ~d ~ pYer~ 1WfU ~--~-~

--

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N A-rR))( M A--TLJ--

~Wamp( b ~d

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cllpoundyy-d P(lJ ~eJ ~ ( orh ) middot

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cry )(2- + f _ 1- o ~ b- b

(

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

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)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

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Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

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(il)

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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1

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

o Ioshy

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- 2i 9A~

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5 9

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3

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aL -

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

o Ioshy

(-04) d l bl-Ja ) ~

QIshy 01-shy ~~

l~ l~)

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

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r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

o Ioshy

(-04) d l bl-Ja ) ~

QIshy 01-shy ~~

l~ l~)

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ltxt~

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(

l shy

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-shy- 35

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+) ltS~ -- 51 ~gt Ztmiddot~-~~middotmiddotmiddotmiddotmiddotmiddotmiddot

- 2i 9A~

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J 4ltr~r ~b~~~~ ~

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5 9

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3

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r bullbull ~-- ~ bullbull ~ bullbullbullbull _ _~ _ -~---

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d ~ YrI1 ~ ~ ~ VY1 t- L

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I

lti)shy

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

o Ioshy

(-04) d l bl-Ja ) ~

QIshy 01-shy ~~

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-shy- 35

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- 2i 9A~

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J 4ltr~r ~b~~~~ ~

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5 9

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

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1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

- ok- ~~ ~~ ~~ ~ ~

A~~) tt CIL)-1shy G- ~

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50 Let pointof concurrent is (h k) Equationofnormal at(ii ylis

1 x-x y-y_~_1=-----=-

xla2 - yl b2

Itpasses through (hk)~then y2a2(h~x)+ b2x2 b4k2x2 (i)

X2 y 12 -1b2

But --+-- = 1 or v =_(a2 _X 2) (ii)

a2 b2 a2bull

Valueofy2 from eq (ii)puttlng in eq (i) we get

b Cal X li)a2h +(b2 - a2)x 2 =b4k2x2 a

=gt b az _ x 12)a4h2 +(bZ _a2)2x 12+2a2hx(b2 _a2 )

a = b4k2xz Arranging above as a~(QUrthdegreeequation in x we get =gt -(a2 ~ b2)2 X4 + 2ha(a2 b2) x 3 + x2(bullbull) ~ 2a4h(a2 ~tf)x +~qh2=O abovetheequation beingoffoutth degree in x thereforero()ts(j~fheaboveeqpatibnare

Xl X2 Xh X4 then ~ha2(a2 - b2 ) 2haZ

(x + 2 + X3+ X4) = - _(a2_b2)2= (a2 _ti2)

( _1 +_1_+_1 +_1__)= EXtX2X 3

Xl X 2 x3 x xt x2 x3 x

2a4h(a2 - b2

)

_(a2 _ b2 ) 2 2(a2 _b2 )

a6h 2 - a2b

_(a2 _ b2 ) 2

()bullbull bullbull III

(il)

Multiplying eqs (iii) and (iv) we get (x +X2+ X3+X4) x(~+_l +_1 +_1)=4 Xl X2 X3 X4

- -

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--=~~=~--(C~ G I

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Tmiddot

(

middot

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