X2 T03 01 ellipse (2011)
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Transcript of X2 T03 01 ellipse (2011)
![Page 1: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/1.jpg)
Conics
![Page 2: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/2.jpg)
ConicsThe locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix)
![Page 3: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/3.jpg)
ConicsThe locus of points whose distance from a fixed point (focus) is a multiple, e, (eccentricity) of its distance from a fixed line (directrix)
![Page 4: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/4.jpg)
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
![Page 5: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/5.jpg)
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 < 1 ellipse
![Page 6: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/6.jpg)
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 < 1 ellipse
e = 1 parabola
![Page 7: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/7.jpg)
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 < 1 ellipse
e = 1 parabola
e > 1 hyperbola
![Page 8: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/8.jpg)
Ellipse (e < 1)y
xa-a
b
-b
AA’
![Page 9: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/9.jpg)
Ellipse (e < 1)y
xa-a
b
-b
AA’S Z
![Page 10: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/10.jpg)
Ellipse (e < 1)y
xa-a
b
-b
AA’S Z
SA = eAZ and SA’ = eA’Z
![Page 11: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/11.jpg)
Ellipse (e < 1)y
xa-a
b
-b
AA’S Z
SA = eAZ and SA’ = eA’Z
(1) SA’ + SA = 2a(2) SA’ – SA = e(A’Z – AZ)
![Page 12: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/12.jpg)
Ellipse (e < 1)y
xa-a
b
-b
AA’S Z
SA = eAZ and SA’ = eA’Z
(1) SA’ + SA = 2a(2) SA’ – SA = e(A’Z – AZ)
= e(AA’)= e(2a)= 2ae
![Page 13: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/13.jpg)
xa-a
b
-b
AA’S Z
(1) + (2); 2SA’ = 2a(1 + e)SA’ = a(1 + e)
![Page 14: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/14.jpg)
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)
![Page 15: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/15.jpg)
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)
![Page 16: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/16.jpg)
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
![Page 17: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/17.jpg)
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)
![Page 18: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/18.jpg)
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
![Page 19: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/19.jpg)
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
![Page 20: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/20.jpg)
xa-a
b
-b
PAA’
S Z
0,aeS
yxP , N
y
eaN ,
![Page 21: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/21.jpg)
xa-a
b
-b
PAA’
S Z
0,aeS
yxP , N
y
eaN ,
ePNSP
![Page 22: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/22.jpg)
xa-a
b
-b
PAA’
S Z
0,aeS
yxP , N
y
eaN ,
ePNSP
2
222
22
22 0
eaxeyaex
yyeaxeyaex
![Page 23: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/23.jpg)
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
![Page 24: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/24.jpg)
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
![Page 25: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/25.jpg)
byx ,0when 222
22
2
1
11
i.e.
eabea
b
![Page 26: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/26.jpg)
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
![Page 27: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/27.jpg)
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
Note: If b > a
foci on the y axis 222 1 eba
be,0:focus
eby :sdirectrice
![Page 28: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/28.jpg)
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
Note: If b > a
foci on the y axis 222 1 eba
be,0:focus
eby :sdirectrice
abArea
![Page 29: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/29.jpg)
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
![Page 30: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/30.jpg)
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
159
22
yx
392
aa
![Page 31: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/31.jpg)
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
159
22
yx
392
aa
52 b 51 22 ea
![Page 32: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/32.jpg)
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
yx
159
22
yx
392
aa
52 b 51 22 ea
3294951
519
2
2
2
e
e
e
e
![Page 33: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/33.jpg)
e.g.
features.
important theof all showing ellipse sketch the and 159
ellipsetheofsdirectriceandfocity,eccentrici theFind22
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
![Page 34: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/34.jpg)
y
x
Auxiliary circle
3-3
![Page 35: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/35.jpg)
y
x
Auxiliary circle
3-3
35
ab
![Page 36: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/36.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
![Page 37: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/37.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
![Page 38: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/38.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
![Page 39: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/39.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
![Page 40: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/40.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
![Page 41: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/41.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
![Page 42: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/42.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
![Page 43: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/43.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
S(2,0)S’(-2,0)
![Page 44: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/44.jpg)
y
x
Auxiliary circle
3-3
35
ab
5
5
S(2,0)S’(-2,0)
29
x 29
x
![Page 45: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/45.jpg)
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
![Page 46: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/46.jpg)
(ii) 011161849 22 yxyx
![Page 47: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/47.jpg)
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
![Page 48: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/48.jpg)
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
94
41
3611
92
41 22
yx
![Page 49: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/49.jpg)
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
94
41
3611
92
41 22
yx
19
241 22
yx
![Page 50: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/50.jpg)
(ii) 011161849 22 yxyx
3611
94
42 22
yyxx
)2,1(:centre
94
41
3611
92
41 22
yx
19
241 22
yx
![Page 51: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/51.jpg)
(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
![Page 52: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/52.jpg)
(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
![Page 53: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/53.jpg)
(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
![Page 54: X2 T03 01 ellipse (2011)](https://reader035.fdocuments.in/reader035/viewer/2022062513/5559ca5ed8b42a98208b4749/html5/thumbnails/54.jpg)
(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