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ELLIPSE
OBJECTIVES
1.
If (-4, 1) and (6, 1) are the vertices of an ellipse and one of its foci lies on the line
x - 2y = 2, then its eccentricity is
a) 2/7
2.
b) 3/7
4.
d) 3/5
The foci of the ellipse 9x2 + 25y2 - 18x - 100y - 116 = 0 is
a) (5,-2), (-3,2)
3.
c) 4/7
b) (5,2)(-3,2)
c) (5,2),(3,2)
d)(-5,-2),(-3,2)
The equations of the directrices of the ellipse 25x2 + 9y2 - 150x - 90y +225 = 0 are
a) 4y+5 = 0, 4y-45 = 0
b) 4y+35 = 0, 4y - 15 = 0
c) 4y+35 = 0, 4y-25 = 0
d) 4x-35 = 0, 4x + 35 = 0
In an ellipse, the distance between the foci is 8 and the distance between the directrices
is 25. Then the length of major axis is
b) 20 2
a) 10 2
5.
7.
d) 40 2
The distance between the foci of 4x2 +y2 - 16x - 6y - 39 = 0 is
a) 3 3
6.
c) 30 2
c) 8 3
c) 8
d) 16
The auxilary circle of the ellipse 9x2 + 25y2-18x-100y-116=0
a) (x+1)2 + (y+2)2 = 9
b) (x+1)2+(y+1)2=25
c) (x-1)2+(y-2)2=9
d) (x-1)2+(y-2)2 = 25
The equation of the locus of the point of intersection of the perpendicular tangents to
the ellipse 9x2 + 16y2 = 144 is
a) x2 + y2 = 5
8.
c) x2 + y2 = 25
d) x2 + y2 = 2
The radius of the director circle of the ellipse 9x2 + 25y2 - 18x - 100y - 116 = 0 is
a) 34
9.
b) x2 + y2 = 7
b) 29
c) 5
d) 8
The focus and the centre of an ellipse are (2, 3), (3, 4). Then the equation of the minor
axis is
a) x + y = 5
10. The equation
(a) A circle
b) x - y +1 = 0
2 x 2 + 3 y 2 = 30
c) x + y - 7 = 0
d) x + y + 7 = 0
represents
(b) An ellipse
(c) A hyperbola
(d) A parabola
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11. The equation of the ellipse whose one of the vertices is (0,7) and the corresponding
directrix is
(a)
y = 12 ,
is
95 x 2 + 144 y 2 = 4655
(b) 144 x 2 + 95 y 2
= 4655
(c)
95 x 2 + 144 y 2 = 13680
(d) None of these
12. The lengths of major and minor axis of an ellipse are 10 and 8 respectively and its major
axis along the y-axis. The equation of the ellipse referred to its centre as origin is
(a)
x2 y2
+
=1
25 16
(b)
x2 y2
+
=1
16
25
13. Equation of the ellipse with eccentricity
(a)
x2 y2
+
=1
3
4
(b)
x2
y2
+
=1
100 64
(c)
x2 y2
+
=1
4
3
1
2
(d)
and foci at
4
x2 y2
+
=
3
4
3
(c)
x2
y2
+
=1
64 100
(±1, 0)
is
(d) None of these
14. The equation of ellipse whose distance between the foci is equal to 8 and distance between
the directrix is 18, is
(a)
5 x 2 − 9 y 2 = 180
(b) 9 x 2 + 5 y 2
= 180
(c)
x 2 + 9 y 2 = 180
(d) 5 x 2 + 9 y 2
= 180
15. The equation of an ellipse whose focus (–1, 1), whose directrix is
eccentricity is
1
2
x −y +3 =0
and whose
, is given by
(a)
7 x 2 + 2 xy + 7 y 2 + 10 x − 10 y + 7 = 0
(b) 7 x 2 − 2 xy + 7 y 2 − 10 x + 10 y + 7 = 0
(c)
7 x 2 − 2 xy + 7 y 2 − 10 x − 10 y − 7 = 0
(d) 7 x 2 − 2 xy + 7 y 2 + 10 x + 10 y − 7 = 0
16. The equation of an ellipse whose eccentricity is 1/2 and the vertices are (4, 0) and (10, 0) is
(a)
3 x 2 + 4 y 2 − 42 x + 120 = 0
(b)
3 x 2 + 4 y 2 + 42 x + 120 = 0
(c)
3 x 2 + 4 y 2 + 42 x − 120 = 0
(d)
3 x 2 + 4 y 2 − 42 x − 120 = 0
17. The eccentricity of the ellipse
(a)
5
14
(b)
25 x 2 + 16 y 2 = 100
4
5
(c)
, is
3
5
(d)
2
5
18. The equation of the ellipse whose one focus is at (4, 0) and whose eccentricity is 4/5, is
(a)
x2
32
+
y2
52
=1
(b)
x2
52
+
y2
32
=1
(c)
x2
52
+
y2
42
=1
(d)
x2
42
+
y2
52
=1
19. The distance between the foci of an ellipse is 16 and eccentricity is
axis of the ellipse is
(a) 8
(b) 64
(c) 16
(d) 32
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1
2
. Length of the major
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20. If the distance between a focus and corresponding directrix of an ellipse be 8 and the
eccentricity be 1/2, then length of the minor axis is
(a) 3
(b)
(c) 6
4 2
21. The length of the latus rectum of the ellipse
(a)
(b)
5 /4
(d) None of these
5 x 2 + 9 y 2 = 45
(c) 5/3
5 /2
is
(d) 10/3
22. Eccentricity of the ellipse whose latus rectum is equal to the distance between two focus points,
is
5 +1
2
(a)
5 −1
2
(b)
(c)
5
2
3
2
(d)
23. If the eccentricity of an ellipse be 5/8 and the distance between its foci be 10, then its latus
rectum is
(a) 39/4
(b) 12
(c) 15
(d)
37/2
24. The equation of the ellipse whose centre is at origin and which passes through the points
(–3, 1) and (2, –2) is
(a)
5 x 2 + 3 y 2 = 32
(b)
3 x 2 + 5 y 2 = 32
(c) 5 x 2 − 3 y 2
= 32
(d) 3 x 2 + 5 y 2 + 32 = 0
25. If the length of the major axis of an ellipse is three times the length of its minor axis, then
its eccentricity is
(a)39/4
(b)12
2 2
3
(d) e =
(c) 15
26. The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between
the foci. The equation of the ellipse is
(a)
x 2 + 2 y 2 = 100
(b)
x2 +
2 y 2 = 10
(c) x 2 − 2 y 2
= 100
(d) None of these
27. The locus of a variable point whose distance from (–2, 0) is
line
x =−
9
2
(a) Ellipse
2
3
times its distance from the
, is
(b) Parabola
(c) Hyperbola
28. The length of the latus rectum of an ellipse is
(a)
2
3
(b)
2
3
(c)
5×4 ×3
73
1
3
(d) None of these
of the major axis. Its eccentricity is
(d)
3
 
4
4
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x2
y2
x2 y2
+
= 1 and 2 + 2 = 1
169
25
a
b
29. If the eccentricity of the two ellipse
a/b
is
(a) 5/13
(b) 6/13
(c) 13/5
30. P is any point on the ellipse
9 x 2 + 36 y 2 = 324
(a) 3
(b) 12
x2 y2
+
=1
36 11
32. The equation
(a)
r>2
(b)
x2
y2
+
=1
6
11
x2
y2
+
+1 = 0
2−r r−5
(b)
(d) 13/6
, whose foci are S and S’. Then
(c) 36
31. The equation of the ellipse whose foci are
(a)
are equal, then the value of
(c)
SP + S ' P
equals
(d) 324
(±5, 0)
and one of its directrix is
x2 y2
+
=1
6
11
5 x = 36
, is
(d) None of these
represents an ellipse, if
2<r<5
(c)
r>5
(d) None of these
33. If the distance between the foci of an ellipse be equal to its minor axis, then its eccentricity
is
(a) 1/2
(b) 1 /
(c)
2
1/3
34. The equations of the directrices of the ellipse
(a)
2 x = ±25
(b)
5 x = ±9
(c)
(d)
16 x 2 + 25 y 2 = 400
3 x = ±10
1/ 3
are
(d) None of these
35. If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is
(a) 3/2
(b)
(c)2/3
3 /2
(d)
3 x 2 + 4 y 2 = 48
36. The distance between the foci of the ellipse
(a) 2
(b) 4
37. The distance of the point
(a)
a(e + cos θ )
(c) 6
' θ ' on
the ellipse
(b) a(e − cos θ )
(c)
2 /3
is
(d) 8
x2
a
2
+
y2
b2
a(1 + e cos θ )
=1
from a focus is
(d) a(1 + 2e cos θ )
38. If a bar of given length moves with its extremities on two fixed straight lines at right
angles, then the locus of any point on bar marked on the bar describes a/an
(a) Circle
(b) Parabola
39. The centre of the ellipse (x + y − 2)
9
(a) (0, 0)
(b) (1, 1)
2
(c)Ellipse
+
(x − y ) 2
=1
16
(c) (1, 0)
(d)Hyperbola
is
(d) (0, 1)
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40. The equation of the ellipse whose centre is (2, –3), one of the foci is (3, –3) and the
corresponding vertex is (4, –3) is
(a)
( x − 2) 2 (y + 3) 2
+
=1
3
4
(b)
( x − 2) 2 (y + 3) 2
+
=1
4
3
41. The length of the axes of the conic
(a)
1
,9
2
(b)
(c)
42. The eccentricity of the conic
3
2
(a)
(b)
x2 y2
+
=1
3
4
9x 2 + 4y 2 − 6x + 4y + 1 = 0
2
5
3,
(c)
1,
2
3
3
4
(c)
is
(d)
3
43. The equation of the tangent to the ellipse
x 2 + 16 y 2 = 16
(c)
3x − y ± 7 = 0
(a)
44. If
(b)
3x − y + 7 = 0
y = mx + c
3x − y −7 = 0
x2 y2
+
=1,
9
4
is tangent on the ellipse
(a) 0
(b)
making an angle of
60 o
with x-axis is
(d) None of these
then the value of c is
± 9m 2 + 4
(c)
3/m
, are
(d) 3, 2
4 x 2 + 16 y 2 − 24 x − 3 y = 1
1
2
(d) None of these
(d)
± 3 1+m2
45. The equation of the tangents drawn at the ends of the major axis of the ellipse
9 x 2 + 5 y 2 − 30 y = 0
(a)
, are
y = ±3
(b)
x=± 5
y = 0, y = 6
(c)
(d) None of these
46. The locus of the point of intersection of mutually perpendicular tangent to the ellipse
x2
a
2
+
y2
b2
=1,
is
(a) A straight line
(b) A parabola
(c)A circle
(d) None of these
47. The equation of normal at the point (0, 3) of the ellipse
(a)
48.
y−3 =0
(b)
y+3 =0
If the foci of an ellipse are
(± 5 , 0)
(c) x-axis
9 x 2 + 5 y 2 = 45
is
(d) y-axis
and its eccentricity is
5
3
, then the equation of the
ellipse is
(a)
9 x 2 + 4 y 2 = 36
(b)
4 x 2 + 9 y 2 = 36
(c)
36 x 2 + 9 y 2 = 4
(d) 9 x 2 + 36 y 2 = 4
49. An ellipse has OB as semi minor axis, F and F′′ its foci and the angle FBF′′ is a right angle.
Then the eccentricity of the ellipse is
(a)
1
4
(b)
1
3
©
1
2
(d) 1
2
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50.
The value of λ , for which the line
3
2
(a)
51.
(b)
1
2
8
λy = −3
3
2x −
(c) −
is a normal to the conic
3
2
4 x 2 + 5y 2 = 1
The sum of the focal distances of any point on the conic
53.
If the line
(b) 9
x cos α + y sin α = p
(c) 41
p 2 (a 2 cos 2 α + b 2 sin 2 α ) = a 2 − b 2
(c)
p 2 (a 2 sec 2 α + b 2 cosec 2α ) = a 2 − b 2
(b)
(d)
54. Eccentric angle of a point on the ellipse
(d)Is focus of the curve
x 2 y2
+
=1
25 16
is
(d)18
x2
be normal to the ellipse
(a)
is
8
(a) Lies on the curve (b) Is inside the curve (c)Is outside the curve
(a) 10
y2
=1
4
(d) 3
The point (4, –3) with respect to the ellipse
52.
x2 +
a
+
2
y2
b2
=1,
then
p 2 (a 2 cos 2 α + b 2 sin 2 α ) = (a 2 − b 2 ) 2
p 2 (a 2 sec 2 α + b 2 cosec 2α ) = (a 2 − b 2 )2
x 2 + 3y 2 = 6
at a distance 2 units from the centre of
the ellipse is
(a)
π
4
(b)
π
(c) 3π
3
(d) 2π
4
55. The equations of the tangents of the ellipse
3
9 x 2 + 16 y 2 = 144
which passes through the point
(2, 3) is
(a)
56.
(b)
y = −3, x − y = 5
(c) y = −4, x + y = 3
(d)
y = −4, x − y = 3
The eccentric angles of the extremities of latus recta of the ellipse
(a)
57.
y = 3, x + y = 5
 ae 
tan −1  ±

 b 
(b)
 be 
tan −1  ±

 a 
(c) tan −1  ±

b 

ae 
In the ellipse, minor axis is 8 and eccentricity is
(a) 6
(b) 12
58.Latus rectum of ellipse
(a) 8/3
(c) 10
4 x 2 + 9 y 2 − 8 x − 36 y + 4 = 0
(b) 4/3
(c)
5
3
(d) tan −1  ±

5
3
x2
a2
+
y2
b2
a 

be 
. Then major axis is
(d)16
is
(d)16/3
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=1
are given by
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59. The sum of focal distances of any point on the ellipse with major and minor axes as 2a
and 2b respectively, is equal to
(a) 2a
(b)
60. The centre of the ellipse
(a) (1,3)
2a
b
(c)
b2
a
(d)
4 x 2 + 9 y 2 − 16 x − 54 y + 61 = 0
(b) (2, 3)
is
(c) (3, 2)
(d) (3, 1)
9 x 2 + 5 y 2 − 18 x − 2 y − 16 = 0
61. The eccentricity of the ellipse
(a) 1/2
2b
a
(b) 2/3
(c) 1/3
is
(d) ¾
62. An ellipse is described by using an endless string which is passed over two pins. If the
axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins
respectively in cm, are
(a)
63. The line
(a)
(b) 6,
6, 2 5
y = mx + c is
(c)
5
a normal to the ellipse
− (2 am + bm 2 )
(b)
(a 2 + b 2 )m
(c)
a2 + b 2m 2
ax
by
−
= a2 − b 2
sin θ cos θ
(b)
ax
by
−
= a2 − b 2
cos θ sin θ
(c)
(d)
65. If the normal at the point
then
cos θ
(a)
P (θ )
x2
a2
+
y2
=1,
a2
if
(a 2 − b 2 )m
−
x2
a
2
+
y2
b2
=1
c=
(d)
a2 + b 2m 2
64. The equation of the normal to the ellipse
(a)
(d) None of these
4, 2 5
(a 2 − b 2 )m
a2 + b 2
at the point
(a cos θ , b sinθ )
ax
by
−
= a2 + b 2
sin θ cos θ
ax
by
−
= a2 + b 2 .
cos θ sin θ
to the ellipse
x2 y2
+
=1
14
5
intersects it again at the point
is equal to
2
3
66. The foci of the ellipse
(b)
−
2
3
is
(c)
3
2
25 (x + 1) 2 + 9(y + 2) 2 = 225
(d)
−
3
2
are at
(a) (–1, 2) and (–1, –6)
(b) (–1, 2) and (6, 1)
(c) (1, –2) and (1, –6)
(d)(–1, –2) and (1, 6 )
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67. If
P ≡ (x , y ) , F1 ≡ (3, 0) , F2 ≡ (−3, 0)
(a) 8
and
16 x 2 + 25 y 2 = 400
(b) 6
PF1 + PF2
, then
(c)10
equals
(d)12
3 x 2 + 2y 2 = 5
68. The angle between the pair of tangents drawn to the ellipse
from the point
(1, 2), is
(a)
 12 
tan −1  
 5 
69. If the line
(a)
70.
(b)
tan −1 (6 5 )
y = mx + c touches
± b 2m 2 + a2
(b)




 5
(c) tan −1  12
the ellipse
x2
b2
± a2m 2 + b 2
+
y2
a2
(c)
=1,
(d) tan −1 (12
then
5)
c=
± b 2m 2 − a2
(d) ±
a2m 2 − b 2
The locus of the point of intersection of the perpendicular tangents to the ellipse
x2 y2
+
=1
9
4
(a)
is
x2 + y2 = 9
(b)
x2 + y2 = 4
(c) x 2 + y 2
= 13
(d)
x2 + y2 = 5
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ELLIPSE
HINTS AND SOLUTIONS
1.
d. 2a=10 and (h+ae,1) is a point on the line.
2.
b
3.
a
4.
a
2ae=8 and 2a/e=25.
5.
b
2ae
6.
d
( x + h) + ( y + k )
7.
c
director circle.
8.
a
9.
c
10.
(b)
11.
(b) Vertex (0,7), directrix
2
2
= a2
minor axis is the line perpendicular to cs passes through c.
x2
y2
x 2 y2
+
=1 ⇒
+
=1.
15 10
(30 / 2) (30 / 3)
Also
b
= 12
e
⇒
e=
y = 12 ,
(b) Here given that
2b = 10, 2a = 8
144 x 2 + 95 y 2 = 4655
⇒
Hence the required equation is
13.
(b) Given that,
e=
⇒ ae = 1 ⇒ a = 2 .
1
2
b =7
7
95
,a=7
12
144
Hence equation of ellipse is
12.
∴
and
Now
.
b = 5, a = 4
x2 y2
+
=1.
16 25
(± ae, 0) = (± 1, 0)
b 2 = a 2 (1 − e 2 )
1

⇒ b 2 = 4 1 −  ⇒ b 2 = 3
4

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Hence, equation of ellipse is
14.
(d)
2a
= 18
e
2ae = 8 ,
e=
⇒
x 2 y2
+
= 1.
4
3
a = 4 ×9 = 6
4
6
2
5 =2 5
, b = 6 1− =
3
9
3
Hence the required equation is
i.e.,
15.
5 x 2 + 9 y 2 = 180
x2 y2
+
=1
36 20
.
(a) Let any point on it be (x,y), then
(x + 1)2 + (y − 1)2
x −y +3
=
1
2
2
Squaring and simplifying, we get
7 x 2 + 2 xy + 7 y 2 + 10 x − 10 y + 7 = 0 .
16.
(a) Major axis
e=
1
2
⇒
b = 3 1−
Equation is
⇒
17.
(c)
= 6 = 2a
1
3 3
=
4
2
. Also centre is (7, 0)
3 x 2 + 4 y 2 − 42 x + 120 = 0 .
∴b > a ,
Here,
therefore
⇒ 4 = 25 (1 − e 2 ) ⇒
4
18.
a=3
(x − 7)2
y2
+
=1
9
(27 / 4 )
x2
y2
+
=1.
4
(25 / 4 )
∴
⇒
e=
(b) Here
Now
3
5
a = 2, b = 5 / 2
a 2 = b 2 (1 − e 2 )
16
= 1 − e2
25
⇒
e2 = 1 −
ae = 4 and e =
4
⇒a=5
5
16 

b 2 = a 2 (1 − e 2 ) ⇒ b 2 = 25  1 −
=9
25 

x 2 y2
+
=1 .
25
9
(d) Given, distance between the foci
= 2ae = 16
that length of the major axis of the ellipse
20.
(d)
,
.
Hence equation of the ellipse is
19.
16
9
=
25 25
a
− ae = 8 .
e
Also
e=
1
2
⇒
a=
and eccentricity of ellipse (e)
= 2a =
2ae
16
=
= 32 .
e
1/2
8e
8 .4 16
=
=
(1 − e 2 ) 2(3) 3
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= 1/2 .
We know
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1  16 3 8 3

=
1 −  =
4
3 2
3

16
3
∴b =
Hence the length of minor axis is
21.
22.
(b)
2b 2
= 2 ae
a
Also
⇒
23.
(a)
a=
2b 2
2 ⋅ 5 10
=
=
a
3
3
b 2 = a 2e
e = 1−
b2
a2
e=
Therefore
=
−1 ± 5
2
2
a
(b)
e=
.
b2
a2
. As
or
e2 + e −1 = 0
e < 1, ∴ e =
x2 y2
+
=1 .
a2 b 2
b2 =
2 × 39 39
=
8
4
=
Since it passes through (–3, 1) and (2, –2), so
32
5
2a = 3(2b ) ⇒ a 2 = 9 b 2 = 9 a 2 (1 − e 2 ) ⇒ e =
(a) Given
Also
⇒
2b 2
= 10
a
and
.
⇒
b 2 = a 2 (1 − e 2 )
b=
a
2
or
b=5 2
(a) Let point P
So,
2 2
3
.
2b = 2ae
,
e 2 = (1 − e 2 )
⇒
e=
1
2
a = 10
Hence equation of ellipse is
27.
3 x 2 + 5 y 2 = 32
(d) Major axis = 3(Minor axis)
⇒
26.
.
.
Hence required equation of ellipse is
25.
5 −1
2
10
25
39
= 8 , b = 8 1−
=8
2 .5
64
8
8
Latus rectum = 2b
24.
or
e2 = 1 − e
or
.
x2 y2
+
=1.
9
5
(d) Here the ellipse is
Latus rectum
16 3
3
x2
y2
+
=1
(10 )2 (5 2 )2
(x 1 , y 1 )
( x 1 + 2)2 + y 12 =
2
9
 x1 + 
3
2
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9
a2
+
1
b2
= 1 and
1
a2
+
1
b2
=
1
4
⇒ a2
=
32
3
,
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⇒
81


+ 9 x1 
9[ x 12 + y12 + 4 x 1 + 4 ] = 4  x 12 +
4


⇒
5 x 12 + 9 y 12 = 45
2
⇒ x1
2b 2 2 a
=
a
3
⇒
+
9
(x1 , y1 )
(b) Latus rectum
⇒
29.
2
(x 1 + 2)2 + y 12 =
Locus of
28.
4
9
 x1 + 
9
2
⇒
is
= 1/3
y 12
=1 ,
5
x2 y2
+
=1,
9
5
which is equation of an ellipse.
(Major axis)
a 2 = 3 b 2 = 3 a 2 (1 − e 2 ) ⇒ e =
.
e = 1 − (25 / 169 )
(c) In the first case, eccentricity
In the second case,
2
3
e ' = 1 − (b 2 / a 2 )
According to the given condition,
1 − b 2 / a 2 = 1 − (25 / 169 )
⇒ b / a = 5 / 13 , (∵ a > 0, b > 0)
⇒
a / b = 13 / 5 .
SP + S ' P = 2a = 2 .6 = 12 .
30.
(b)
31.
(a) Foci
So,
(±5, 0) ≡ (±ae, 0) .
a 36
=
, ae = 5
e
5
Therefore,
⇒
b = 6 1−
Hence equation is
32.
(b)
a=6
36 
a

x =
≡x=
5 
e

and
e=
5
6
25
11
=6
= 11
36
6
x2 y2
+
=1.
36 11
x2
y2
x2
y2
+
+1 = 0 ⇒
+
=1
2−r r−5
r−2 5 −r
Hence
33.
Directrix
r > 2 and r < 5 ⇒ 2 < r < 5 .
(b) Foci are
Also,
(±ae ,0 ) .
Therefore according to the condition,
b 2 = a 2 (1 − e 2 ) ⇒ e 2 = (1 − e 2 ) ⇒ e =
1
2
2ae = 2b or ae = b
.
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34.
(d)
x 2 y2
+
=1
25 16
⇒
e = 1−
16
3
=
25 5
Therefore, directrices are
35.
(b)
2b 2
=b
a
Hence
36.
(b)
⇒
e = 1−
b 1
=
a 2
⇒
b2
3
2
=
a2
a2
=
5
=0
3/5
or
3 x ± 25 = 0
.
1
4
.
x2
y2
+
=1
(48 / 3) (48 / 4 )
a 2 = 16 , b 2 = 12
⇒
e = 1−
Distance is
37.
b2
x±
b2
1
=
2
a2
2ae = 2 ⋅ 4 ⋅
1
=4.
2
(c) Focal distance of any point P (x,y) on the ellipse is equal to
Here
SP = a + ex
. Here
x = a cos θ
SP = a + ae cos θ = a(1 + e cos θ ) .
38.
(c) It is obvious.
39.
(b) The centre of the given ellipse is the point of intersection of the lines
x − y = 0 i.e.,(1,1).
40.
(b) Foci
= (3,−3) ⇒ ae = 3 − 2 = 1
Vertex
⇒
= (4 ,−3) ⇒ a = 4 − 2 = 2 ⇒ e =
1 2

b = a 1 −  =
4 2

1
2
3 = 3
Therefore, equation of ellipse with centre
(2,−3) is
( x − 2)2 (y + 3)2
+
=1.
4
3
41.
(c) Given that, the equation of conic
9x 2 + 4y 2 − 6x + 4y + 1 = 0
2
⇒ (3 x − 1) 2 + (2 y + 1)2
Here
a=
1
3
,
b=
1
2
=1⇒
;∴
Length of axes are
2a =
1

x − 
3
(y + 1) 2

+
=1 .
1
1
9
2
2
3
,
2b = 1.
 2
 1,  .
 3
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x +y −2 =0
and
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42.
4 x 2 + 16 y 2 − 24 x − 3 y = 1
(a) Given equation of conic is
⇒ (2 x − 6 )2 + (4 y − 4 )2 = 53
⇒ 4 ( x − 3 )2 + 16 (y − 1)2 = 53
⇒
( x − 3)2 (y − 1)2
+
=1
53 / 4
53 / 16
∴ e = 1−
m = tan 60 ° = 3
43.
(c)
44.
(c) Here,
∴
45.
(c)
b2
53 / 16
1
3
= 1−
= 1− =
2
53 / 4
4
2
a
. Tangent is
a = 3, b = 2 .
c 2 = 4 + 9m 2 ;
y=
c = ± 9m 2 + 4
⇒
y=
3x ±7
.
c 2 = b 2 + a 2m 2
.
9 x 2 + 5(y 2 − 6 y ) = 0
⇒
9 x 2 + 5(y 2 − 6 y + 9) = 45
∵ a2 < b 2,
Then
⇒x
2
5
(y − 3)2
=1
9
+
so axis of ellipse on y-axis.
At y axis, put
x =0,
so we can obtained vertex.
0 + 5 y 2 − 30 y = 0 ⇒ y = 0, y = 6
Therefore, tangents of vertex
46.
(c) It is a fundamental concept.
47.
(d) For
x2 y2
+
= 1,
a2 b 2
⇒ (x − x1 )a
x1
⇒
48.
3 x ± 1 + 3 ⋅ 16
∴ By formula,
∴
.
2
=
equation of normal at point
(y − y1 )b 2
y1
( x − 0)
(y − 3).9
5=
0
3
(b) ∵ ae = ±
5
⇒
y = 0, y = 6 .
(x 1 , y 1 ) ,
; ∴ (x1 , y1 ) ≡ (0,3), a 2 = 5, b 2 = 9
or
x =0
 3
a = ± 5 
 5

 = ±3

⇒
a2 = 9
∴ b 2 = a 2 (1 − e 2 ) = 9  1 − 5  = 4

9
Hence, equation of ellipse
49.
(c)
x2 y2
+
=1
9
4
∠F′BF = 90 ° , F′B⊥FB
i.e., slope of
⇒
(F′B)
× Slope of
b
b
×
= −1 , b 2 = a 2 e 2
ae − ae
(F′B) = −1
…..(i)
Y
B
X′
X
F′ O
F
Y′
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We know that
e = 1−
e 2 = 1 − e 2 , 2e 2 = 1 , e 2 =
50.
b2
a2e 2
=
1
−
= 1 − e2
a2
a2
1
2
,
1
e=
2
.
(d) We know that the equation of the normal at point
(a cos θ , b sinθ )
on the curve
x2 +
y2
=1
4
is
given by
ax sin θ − by cosec θ = a 2 − b 2
.....(i)
Comparing equation (i) with
a sin θ = 2 , b cosec θ =
∵a = 1, b = 2 ;
∴ 2 = 16 λ or
3
or
16
λ
3
ab =
We get,
.....(ii)
λ = 3 /8
x2
y2
+
=1
1/4 1/5
51. (c) Given ellipse is
∴
8
λ
3
8
λy = −3 .
3
2x −
16
9
+
− 1 = 64 + 45 − 1 > 0
1/4 1/5
Point (4, –3) lies outside the ellipse.
52. (a) For any point P on the ellipse have focus S and S′
∴ For
SP + S ′P = 2a
x 2 y2
+
=1
25 16
Sum of focal distances = 2 × 5 = 10.
53. (d) The equation of any normal is
Given line
x cos α + y sin α = p
ax sec φ − by cosec φ = a 2 − b 2
.....(i)
comparing and eliminating φ
⇒ p 2 (b 2 cosec 2 α + a 2 sec 2 α ) = (a 2 − b 2 )2 .
54.
(a ) Let the eccentric angle of the point be θ , then its
Hence
Hence,
55.
6 cos 2 θ + 2 sin2 θ = 4 or cos 2 θ =
cos θ = ±
1
2
, ∴ θ = π or
(a) The tangent will be
4
3π
4
y − 3 = m ( x − 2)
( 6 cos θ , 2 sin θ ).
1
2
.
⇒
y − mx = 3 − 2m .
But it is tangent to the given ellipse, therefore
56.
co-ordinates are
(c) Coordinates of any point on the ellipse
m = 0, − 1 .
Hence tangents are
x2 y2
+
=1
a2 b 2
(a cos θ , b sin θ ).
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y = 3 and x + y = 5 .
whose eccentric angle is
θ
are
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The coordinates of the end points of latus recta are
⇒
57.
tan θ = ±
e2 =1 −
b2
or
a2
axis of the ellipse
58.
and
b sin θ = ±
b2
a
b
 b 
⇒ θ = tan − 1  ±
.
ae
 ae 
(2 b) = 8
(b) Given, minor axis of ellipse
ellipse,
2 

 ae , ± b . ∴ a cos θ = ae

a 

(a) The ellipse is
5
16
=1 − 2
9
a
or
16
or
=1 −
a2
b=4
and eccentricity
5
9
a2 =
or
16 × 9
= 36
4
or
(e ) = 5 / 3
a = 6.
. We know that in an
We also know that major
= 2a = 2 × 6 = 12 .
4 (x − 1)2 + 9(y − 2)2 = 36
Therefore, latus rectum = 2b
a
2
=
2.4 8
=
3
3
.
59. (a) Sum of focal distances of a point in an ellipse is always equal to length of major axis of that
ellipse. It is a property of ellipse.
60.
(b) 4 (x − 2)2 + 9(y − 3)2 = 36
Hence the centre is (2, 3).
61.
(b) Given equation can be written as
( x − 1) 2 (y − 2) 2
+
=1 ; e =
5
9
62.
(d) Given
e2 = 1 −
b 2 − a2
b
2
9−5 2
=
9
3
.
2a = 6, 2b = 4 i.e. , a = 3, b = 2
b2 5
=
a2 9
⇒e =
5
3
= 2 ae = 2 5 cm
Distance between the pins
Length of string
63.
=
= 2 a + 2 ae = 6 + 2 5 cm
(c) As we know that the line
.
lx + my + n = 0
is normal to
x2 y2
+
=1 ,
a2 b 2
if
a2 b 2
(a 2 − b 2 )2
+
=
l2 m 2
n2
. But in this
condition, we have to replace l by m, m by –1 and n by c, then the required condition is
c=±
(a 2 − b 2 )m
a 2 + b 2m 2
.
ax sec θ − by cosec θ = a 2 − b 2 .
64.
(c)
65.
(b) The normal at
P(a cos θ , b sin θ )
It meets the curve again at
∴
is
ax sec θ − by cosec θ = a 2 − b 2 ,
Q(2θ )
where
i.e., (a cos 2θ , b sin 2θ ) .
a
b
a cos 2θ −
(b sin 2θ ) = a 2 − b 2
cos θ
sin θ
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a 2 = 14 , b 2 = 5
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⇒
14
5
cos 2θ −
(sin 2θ ) = 14 − 5
cos θ
sin θ
⇒ 18 cos 2 θ − 9 cos θ − 14 = 0
⇒ (6 cos θ − 7)(3 cos θ + 2) = 0 ⇒ cos θ = −
66.
(a)
2
3
.
(x + 1)2 (y + 2)2
+
=1
225
225
25
9
a=
225 15
=
,b =
25
5
⇒
225 15
=
9
3
e = 1−
9
4
=
25
5
Focus =  − 1,−2 ± 15 . 4  = (−1,−2 ± 4 ) =(–1,2); (–1,–6) .

67.
3 5
(c) Equation of the curve is
x2 y2
+
=1
52 42
⇒ −5 ≤ x ≤ 5, − 4 ≤ y ≤ 4
PF1 + PF2 = [(x − 3)2 + y 2 ] + [(x + 3)2 + y 2 ]
= ( x − 3 )2 +
400 − 16 x 2
400 − 16 x 2
+ ( x + 3) 2 +
25
25
=
1
2
2

 (9 x + 625 − 150 x ) + (9 x + 625 + 150 x ) 
5

=
1
1
2
2 
 (3 x − 25 ) + (3 x + 25 )  = {25 − 3 x + 3 x + 25 }
 5
5
= 10
68.
(c)
SS 1 = T 2
tan θ = 2
h 2 − ab
, a = 9 , b = −4
a+b
and
h = −12 .
69.
(a) Since, here a and b are interchanged.
70.
(c) The locus of point of intersection of two perpendicular tangents drawn on the ellipse is
x 2 + y 2 = a2 + b 2,
which is called ‘director- circle’.
Given ellipse is
x2 y2
+
=1,
9
4
∴Locus is
x 2 + y 2 = 13 .
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