X2 t03 01 ellipse (2012)

Conics

ConicsThe locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix)


e = 0 circle


e = 0 circle

e < 1 ellipse


e = 0 circle

e < 1 ellipse

e = 1 parabola


e = 0 circle

e < 1 ellipse

e = 1 parabola

e > 1 hyperbola

Ellipse (e < 1)y

xa-a

b

-b

AA’

Ellipse (e < 1)y

xa-a

b

-b

AA’S Z

Ellipse (e < 1)y

xa-a

b

-b

AA’S Z

SA = eAZ and SA’ = eA’Z

Ellipse (e < 1)y

xa-a

b

-b

AA’S Z


(1) SA’ + SA = 2a(2) SA’ – SA = e(A’Z – AZ)

Ellipse (e < 1)y

xa-a

b

-b

AA’S Z


(1) SA’ + SA = 2a(2) SA’ – SA = e(A’Z – AZ)

= e(AA’)= e(2a)= 2ae

xa-a

b

-b

AA’S Z

(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)

xa-a

b

-b

AA’S Z

(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)

(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)

xa-a

b

-b

AA’S Z

OS = OA - SA

(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)

Focus

(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)

xa-a

b

-b

AA’S Z

OS = OA - SA

(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)

Focus

= a – a(1 – e)= ae

(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)

0,aeS

xa-a

b

-b

AA’S Z

OS = OA - SA

(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)

Focus

= a – a(1 – e)= ae 0,aeS

OZ = OA + AZ

Directrix

(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)

xa-a

b

-b

AA’S Z

OS = OA - SA

(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)

Focus

= a – a(1 – e)= ae 0,aeS

OZ = OA + AZ

Directrix

eSAOA

(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)

SA eAZ

xa-a

b

-b

AA’S Z

OS = OA - SA

(1) - (2); 2SA = 2a(1 - e)SA = a(1 - e)

Focus

= a – a(1 – e)= ae 0,aeS

OZ = OA + AZ

Directrix

eax sdirectrice

eSAOA

(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)

ea

eea

eae

1

SA eAZ

xa-a

b

-b

PAA’

S Z

0,aeS

yxP , N

y

eaN ,

xa-a

b

-b

PAA’

S Z

0,aeS

yxP , N

y

eaN ,

ePNSP

xa-a

b

-b

PAA’

S Z

0,aeS

yxP , N

y

eaN ,

ePNSP

2

222

22

22 0

eaxeyaex

yyeaxeyaex

xa-a

b

-b

PAA’

S Z

0,aeS

yxP , N

y

eaN ,

ePNSP

2

222

22

22 0

eaxeyaex

yyeaxeyaex

22222

2222222

1122eayex

aaexxeyeaaexx

xa-a

b

-b

PAA’

S Z

0,aeS

yxP , N

y

eaN ,

ePNSP

2

222

22

22 0

eaxeyaex

yyeaxeyaex

22222

2222222

1122eayex

aaexxeyeaaexx

11 22

2

2

2

ea

yax

byx ,0when 222

22

2

1

11

i.e.

eabea

b

byx ,0when 222

22

2

1

11

i.e.

eabea

b

Ellipse: (a > b) 12

2

2

2

by

ax

222 1 where; eab

0,:focus ae

eax :sdirectrice

e is the eccentricity

major semi-axis = a units

minor semi-axis = b units

byx ,0when 222

22

2

1

11

i.e.

eabea

b

Ellipse: (a > b) 12

2

2

2

by

ax

222 1 where; eab

0,:focus ae

eax :sdirectrice




Note: If b > a

foci on the y axis 222 1 eba

be,0:focus

eby :sdirectrice

byx ,0when 222

22

2

1

11

i.e.

eabea

b

Ellipse: (a > b) 12

2

2

2

by

ax

222 1 where; eab

0,:focus ae

eax :sdirectrice




Note: If b > a

foci on the y axis 222 1 eba

be,0:focus

eby :sdirectrice

abArea

e.g.

features.

important theof all showing ellipse sketch the and 159

ellipsetheofsdirectriceandfocity,eccentrici theFind22

yx

e.g.

features.



yx

159

22

yx

392

aa

e.g.

features.



yx

159

22

yx

392

aa

52 b 51 22 ea

e.g.

features.



yx

159

22

yx

392

aa

52 b 51 22 ea

3294951

519

2

2

2

e

e

e

e

e.g.

features.



yx

159

22

yx

392

aa

52 b

0,2:foci

32ty eccentrici

51 22 ea

3294951

519

2

2

2

e

e

e

e

29

233:sdirectrice

x

x

y

x

Auxiliary circle

3-3

y

x

Auxiliary circle

3-3

35

ab

y

x

Auxiliary circle

3-3

35

ab

5

5

y

x

Auxiliary circle

3-3

35

ab

5

5

S(2,0)S’(-2,0)

y

x

Auxiliary circle

3-3

35

ab

5

5

S(2,0)S’(-2,0)

29

x 29

x

y

x

Auxiliary circle

3-3

35

ab

S(2,0)S’(-2,0)

5

5

29

x29

x

Major axis = 6 units unitsaxisMinor 52

(ii) 011161849 22 yxyx

(ii) 011161849 22 yxyx

3611

94

42 22

yyxx

(ii) 011161849 22 yxyx

3611

94

42 22

yyxx

94

41

3611

92

41 22

yx

(ii) 011161849 22 yxyx

3611

94

42 22

yyxx

94

41

3611

92

41 22

yx

19

241 22

yx

(ii) 011161849 22 yxyx

3611

94

42 22

yyxx

)2,1(:centre

94

41

3611

92

41 22

yx

19

241 22

yx

(ii) 011161849 22 yxyx

3611

94

42 22

yyxx

)2,1(:centre

392

bb

94

41

3611

92

41 22

yx

19

241 22

yx

(ii) 011161849 22 yxyx

3611

94

42 22

yyxx

)2,1(:centre

392

bb

35194

12

222

e

eeba

94

41

3611

92

41 22

yx

19

241 22

yx

(ii) 011161849 22 yxyx

3611

94

42 22

yyxx

)2,1(:centre

392

bb

35194

12

222

e

eeba

52,1 :foci 5

92 :sdirectrice y

94

41

3611

92

41 22

yx

19

241 22

yx

(ii) 011161849 22 yxyx

3611

94

42 22

yyxx

)2,1(:centre

392

bb

35194

12

222

e

eeba

52,1 :foci 5

92 :sdirectrice y

94

41

3611

92

41 22

yx

19

241 22

yx Exercise 6A; 1, 2, 3, 5, 7,

8, 9, 11, 13, 15

X2 t03 01 ellipse (2012)

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Transcript of X2 t03 01 ellipse (2012)