Practice Test: Conic Sections, Class 11, Mathematics JEE Notes | EduRev

JEE : Practice Test: Conic Sections, Class 11, Mathematics JEE Notes | EduRev

 Page 1


 
( ) ( ) ( ) ( ) a y b y c y d x = = - = - =
5
4
5
4
3
4
5
4
( ) ( ) ( ) ( ) a a b a c a d a 4 3 8 3 16 3 2 3
CONIC  SEECTION  (TEST)      R1  :  + 1                      Date  :  3
rd
 Jan. 08 
 
1. The coordinates of the point on the parabola y
2
 = 8x, where the ordinate is double the abscissa is 
(a)  (– 2, – 4)  (b)  (2, 4)   (c)  (– 2, 4)   (d)  (2, – 4)
 
2. The equation of the parabola with focus at S (3, 2) and with the line x = – 4 as directrix is 
(a)  y
2
 – 6y – 12 + 3 = 0 (b)  y
2
 – 6y – 12x – 3 = 0            (c)  y
2
 – 6y + 12x + 3 = 0          (d)  y
2
 – 6y + 12x + 3 = 0 
3. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How high is it 2 m 
from the centre. 
(a)  6.3 m   (b)  3.6 m  (c)  4.4 m  (d)  none of these 
4. The equation of the directrix of the parabola x
2
 – 4x – 3y + 10 = 0 is 
 
 
5. The equation of the parabola whose vertex and focus lie on the axis of x at distance a and a
1
 from the origin respectively, is 
(a)  y
2
 = 4 (a
1
 – a) x  (b)  y
2
 = 4 (a
1
 – a) (x – a)  (c)  y
2
 = 4 (a
1
 – a) (x – a
1
)  (d)  none of these 
6. If the focus and vertex of a parabola are the points (0, 2) and (0, 4) respectively, then its equation is 
(a)  x
2
 + 8y = 32  (b)  y
2
 = – 8x + 32  (c)  y
2
 = 8x + 32   (d)  x
2
 – 8y = 32 
7. If the vertex of a parabola is the point (– 3, 0) and the directrix is the line x + 5 = 0, then its equation is 
(a)  y
2
 = 8 (x + 3)  (b)  x
2
 = 8 (y + 3)   (c)  y
2
 = – 8 (x + 3)  (d)  y
2
 = 8 (x + 5) 
8. A variable circle is described to pass through (a, 0) and touch the line x + y = 0. Let S = 0 represent the locus of the centre of 
the circle. Then S = 0 represents 
(a)  an ellipse (b)  a parabola  (c)  a hyperbola             (d)  pair of parallel straight lines 
9. If (2, 0) is the vertex and y–axis the directrix of a parabola then its focus is 
(a)  (4, 0)   (b)  (– 2, 0)   (c)  (2, 0)             (d)  (– 4, 0) 
10. Locus of the middle points of all chords of the parabola y
2
 = 4x which are drawn through the vertex is 
(a)  y
2
 = 2x                                  (b)  y
2
 = 8x                                  (c)  x
2
 + 4y
2
 = 16                                  (d)  x
2
 = 2y 
11. The focus of the parabola y
2
 – x – 2y + 2 = 0 is 
(a)  (1/4, 0)   (b)  (1, 2)  (c)  (5/4, 1)   (d)  (3/4, 1) 
12. The curve represented by x = 3 (cos t + sin t), y = 4 (cos t – sin t) is 
(a)  Ellipse                                      (b)  Parabola                                      (c)  Hyperbola                                       (d)  Circle 
13. The angle made by a double ordinate of length 8a at the vertex of the parabola y
2
 = 4ax is 
(a)  p/2   (b)  p/3   (c)  p/4   (d)  p/6 
14. An equilateral triangle is inscribed in the parabola y
2
 = 4ax whose one vertex is at the vertex of the parabola. The length of its 
side is 
 
 
15. The coordinates of a point on the parabola y
2
 = 8x whose focal distance is 4, are 
(a)  (1/2, ± 2)  (b)  (2, ± 4)  (c)  (1, ± 2 v2)   (d)  none of these 
 
GIITJEE  (GURUS FOR IITJEE)  SCO 382, Sector 37–D, Chandigarh, Ph. 0172–2628810, 2628811, 2628820                         P  –  1 
Page 2


 
( ) ( ) ( ) ( ) a y b y c y d x = = - = - =
5
4
5
4
3
4
5
4
( ) ( ) ( ) ( ) a a b a c a d a 4 3 8 3 16 3 2 3
CONIC  SEECTION  (TEST)      R1  :  + 1                      Date  :  3
rd
 Jan. 08 
 
1. The coordinates of the point on the parabola y
2
 = 8x, where the ordinate is double the abscissa is 
(a)  (– 2, – 4)  (b)  (2, 4)   (c)  (– 2, 4)   (d)  (2, – 4)
 
2. The equation of the parabola with focus at S (3, 2) and with the line x = – 4 as directrix is 
(a)  y
2
 – 6y – 12 + 3 = 0 (b)  y
2
 – 6y – 12x – 3 = 0            (c)  y
2
 – 6y + 12x + 3 = 0          (d)  y
2
 – 6y + 12x + 3 = 0 
3. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How high is it 2 m 
from the centre. 
(a)  6.3 m   (b)  3.6 m  (c)  4.4 m  (d)  none of these 
4. The equation of the directrix of the parabola x
2
 – 4x – 3y + 10 = 0 is 
 
 
5. The equation of the parabola whose vertex and focus lie on the axis of x at distance a and a
1
 from the origin respectively, is 
(a)  y
2
 = 4 (a
1
 – a) x  (b)  y
2
 = 4 (a
1
 – a) (x – a)  (c)  y
2
 = 4 (a
1
 – a) (x – a
1
)  (d)  none of these 
6. If the focus and vertex of a parabola are the points (0, 2) and (0, 4) respectively, then its equation is 
(a)  x
2
 + 8y = 32  (b)  y
2
 = – 8x + 32  (c)  y
2
 = 8x + 32   (d)  x
2
 – 8y = 32 
7. If the vertex of a parabola is the point (– 3, 0) and the directrix is the line x + 5 = 0, then its equation is 
(a)  y
2
 = 8 (x + 3)  (b)  x
2
 = 8 (y + 3)   (c)  y
2
 = – 8 (x + 3)  (d)  y
2
 = 8 (x + 5) 
8. A variable circle is described to pass through (a, 0) and touch the line x + y = 0. Let S = 0 represent the locus of the centre of 
the circle. Then S = 0 represents 
(a)  an ellipse (b)  a parabola  (c)  a hyperbola             (d)  pair of parallel straight lines 
9. If (2, 0) is the vertex and y–axis the directrix of a parabola then its focus is 
(a)  (4, 0)   (b)  (– 2, 0)   (c)  (2, 0)             (d)  (– 4, 0) 
10. Locus of the middle points of all chords of the parabola y
2
 = 4x which are drawn through the vertex is 
(a)  y
2
 = 2x                                  (b)  y
2
 = 8x                                  (c)  x
2
 + 4y
2
 = 16                                  (d)  x
2
 = 2y 
11. The focus of the parabola y
2
 – x – 2y + 2 = 0 is 
(a)  (1/4, 0)   (b)  (1, 2)  (c)  (5/4, 1)   (d)  (3/4, 1) 
12. The curve represented by x = 3 (cos t + sin t), y = 4 (cos t – sin t) is 
(a)  Ellipse                                      (b)  Parabola                                      (c)  Hyperbola                                       (d)  Circle 
13. The angle made by a double ordinate of length 8a at the vertex of the parabola y
2
 = 4ax is 
(a)  p/2   (b)  p/3   (c)  p/4   (d)  p/6 
14. An equilateral triangle is inscribed in the parabola y
2
 = 4ax whose one vertex is at the vertex of the parabola. The length of its 
side is 
 
 
15. The coordinates of a point on the parabola y
2
 = 8x whose focal distance is 4, are 
(a)  (1/2, ± 2)  (b)  (2, ± 4)  (c)  (1, ± 2 v2)   (d)  none of these 
 
GIITJEE  (GURUS FOR IITJEE)  SCO 382, Sector 37–D, Chandigarh, Ph. 0172–2628810, 2628811, 2628820                         P  –  1 
 
x
y
is
2
2
100 49
1 - =
( ) / ( ) / ( ) / ( ) a b c d none of these 17 20 13 20 11 20
( ) , ( ) , ( ) , ( ) , a
a a
b
a a
c
a a
d
a a
4 2 2 4 2 4 4 2
F
H
I
K
-
F
H
I
K
F
H
I
K
-
F
H
I
K
( ) ( ) ( ) ( ) a e b e c e d None of these = = =
2
3
3
2
1
2
( ) / ( ) ( / ) ( ) / ( ) ( / ) a b c x x d 2 3 2 3 5 4 3 7 3 4
3 4
The foci of the ellipse   and the hyperbola coincide, then the value of b
2 x
y
b
x
y
is
a b c d
2
2
2
2
2
16
1
144 81
1
25
7 5 1 9
+ = - =
( ) ( ) ( ) ( )
 
 
 
16. The value of m for which y = mx + 6 is a tangent to the hyperbola   
 
 
17. If y
1
, y
2
 are the ordinates of two points P and Q on the parabola and y
3
 is the ordinate of the point of intersection of tangents 
at P and Q, then 
(a)  y
1
, y
2
, y
3
 are in A.P.               (b)  y
1
, y
3
, y
2
 are in A.P.               (c)  y
1
, y
2
, y
3
 are in G.P.            (d)  y
1
, y
3
, y
2
 are in 
G.P. 
18. The locus of the middle points of the focal chord of the parabola y
2
 = 4ax is 
(a)  y
2
 = 2a (x – a)  (b)  y
2
 = a (x – a)   (c)  y
2
 = 4a (x – a)  (d)  none of these 
19. The length of a local chord of the parabola y
2
 = 4ax making an angle ? with the axis of the parabola is 
(a)  a cosec
2
 ?  (b)  4a sec
2
 ?  (c)  4a cosec
2
 ?  (d)  None of these 
20. If the tangents at P and Q on a parabola meet in T, then SP, ST and SQ are in 
(a)  G.P.                      (b)  A.P.                     (c)  H.P.               (d)  None of these 
21. The point on the curve y
2
 = ax, the tangent at which makes an angle of 45° with x–axis will be given by 
 
 
22. If 2x + y + k = 0 is a normal to the parabola y
2
 = – 8x, then the value of k is 
(a)  – 16                                         (b)  – 8                                         (c)  24                                         (d)  – 24 
23. The equation of the ellipse having vertices at ( ± 5, 0), foci at ( ± 4, 0) is 
(a)  9x
2
 – 25y
2
 – 225 = 0     (b)  9x
2
 + 25y
2
 + 225 = 0       (c)  9x
2
 + 25y
2
 = 225  (d)  None of these 
24. The equation of the locus of all points the sum of whose distances from (3, 0) and (9, 0) is 12 is 
(a)  3x
2
 + 4y
2
 – 36x = 0     (b)  3x
2
 – 4y
2
 – 36x = 0               (c)  3x
2
 + 4y
2 
+ 36x = 0            (d)  3x
2
 + 36y
2
 – 4y = 0 
25. The eccentricity of the ellipse if its latus rectum is equal to one–half of its minor axis is 
 
 
26. The latus rectum of the ellipse 5x
2
 + 9y
2
 = 45 is 
(a)  10/3   (b)  5/3   (c)  2 v5 / 3   (d)  v5 / 3 
27. The length of the latus return of an ellipse is 1/3 of the major axis. Its eccentricity is 
 
 
28.  
 
 
29. The normals at three points P, Q, R of the parabola y
2
 = 4ax meet in (h, k). The centroid of triangle PQR lies on 
(a)  y = 0   (b)  x = 0   (c)  x = – a   (d)  y = a 
30. The equation of the tangent at the vertex of the parabola x
2
 + 4x + 2y = 0 is 
GIITJEE  (GURUS FOR IITJEE)  SCO 382, Sector 37–D, Chandigarh, Ph. 0172–2628810, 2628811, 2628820                         P  –  2 
Page 3


 
( ) ( ) ( ) ( ) a y b y c y d x = = - = - =
5
4
5
4
3
4
5
4
( ) ( ) ( ) ( ) a a b a c a d a 4 3 8 3 16 3 2 3
CONIC  SEECTION  (TEST)      R1  :  + 1                      Date  :  3
rd
 Jan. 08 
 
1. The coordinates of the point on the parabola y
2
 = 8x, where the ordinate is double the abscissa is 
(a)  (– 2, – 4)  (b)  (2, 4)   (c)  (– 2, 4)   (d)  (2, – 4)
 
2. The equation of the parabola with focus at S (3, 2) and with the line x = – 4 as directrix is 
(a)  y
2
 – 6y – 12 + 3 = 0 (b)  y
2
 – 6y – 12x – 3 = 0            (c)  y
2
 – 6y + 12x + 3 = 0          (d)  y
2
 – 6y + 12x + 3 = 0 
3. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How high is it 2 m 
from the centre. 
(a)  6.3 m   (b)  3.6 m  (c)  4.4 m  (d)  none of these 
4. The equation of the directrix of the parabola x
2
 – 4x – 3y + 10 = 0 is 
 
 
5. The equation of the parabola whose vertex and focus lie on the axis of x at distance a and a
1
 from the origin respectively, is 
(a)  y
2
 = 4 (a
1
 – a) x  (b)  y
2
 = 4 (a
1
 – a) (x – a)  (c)  y
2
 = 4 (a
1
 – a) (x – a
1
)  (d)  none of these 
6. If the focus and vertex of a parabola are the points (0, 2) and (0, 4) respectively, then its equation is 
(a)  x
2
 + 8y = 32  (b)  y
2
 = – 8x + 32  (c)  y
2
 = 8x + 32   (d)  x
2
 – 8y = 32 
7. If the vertex of a parabola is the point (– 3, 0) and the directrix is the line x + 5 = 0, then its equation is 
(a)  y
2
 = 8 (x + 3)  (b)  x
2
 = 8 (y + 3)   (c)  y
2
 = – 8 (x + 3)  (d)  y
2
 = 8 (x + 5) 
8. A variable circle is described to pass through (a, 0) and touch the line x + y = 0. Let S = 0 represent the locus of the centre of 
the circle. Then S = 0 represents 
(a)  an ellipse (b)  a parabola  (c)  a hyperbola             (d)  pair of parallel straight lines 
9. If (2, 0) is the vertex and y–axis the directrix of a parabola then its focus is 
(a)  (4, 0)   (b)  (– 2, 0)   (c)  (2, 0)             (d)  (– 4, 0) 
10. Locus of the middle points of all chords of the parabola y
2
 = 4x which are drawn through the vertex is 
(a)  y
2
 = 2x                                  (b)  y
2
 = 8x                                  (c)  x
2
 + 4y
2
 = 16                                  (d)  x
2
 = 2y 
11. The focus of the parabola y
2
 – x – 2y + 2 = 0 is 
(a)  (1/4, 0)   (b)  (1, 2)  (c)  (5/4, 1)   (d)  (3/4, 1) 
12. The curve represented by x = 3 (cos t + sin t), y = 4 (cos t – sin t) is 
(a)  Ellipse                                      (b)  Parabola                                      (c)  Hyperbola                                       (d)  Circle 
13. The angle made by a double ordinate of length 8a at the vertex of the parabola y
2
 = 4ax is 
(a)  p/2   (b)  p/3   (c)  p/4   (d)  p/6 
14. An equilateral triangle is inscribed in the parabola y
2
 = 4ax whose one vertex is at the vertex of the parabola. The length of its 
side is 
 
 
15. The coordinates of a point on the parabola y
2
 = 8x whose focal distance is 4, are 
(a)  (1/2, ± 2)  (b)  (2, ± 4)  (c)  (1, ± 2 v2)   (d)  none of these 
 
GIITJEE  (GURUS FOR IITJEE)  SCO 382, Sector 37–D, Chandigarh, Ph. 0172–2628810, 2628811, 2628820                         P  –  1 
 
x
y
is
2
2
100 49
1 - =
( ) / ( ) / ( ) / ( ) a b c d none of these 17 20 13 20 11 20
( ) , ( ) , ( ) , ( ) , a
a a
b
a a
c
a a
d
a a
4 2 2 4 2 4 4 2
F
H
I
K
-
F
H
I
K
F
H
I
K
-
F
H
I
K
( ) ( ) ( ) ( ) a e b e c e d None of these = = =
2
3
3
2
1
2
( ) / ( ) ( / ) ( ) / ( ) ( / ) a b c x x d 2 3 2 3 5 4 3 7 3 4
3 4
The foci of the ellipse   and the hyperbola coincide, then the value of b
2 x
y
b
x
y
is
a b c d
2
2
2
2
2
16
1
144 81
1
25
7 5 1 9
+ = - =
( ) ( ) ( ) ( )
 
 
 
16. The value of m for which y = mx + 6 is a tangent to the hyperbola   
 
 
17. If y
1
, y
2
 are the ordinates of two points P and Q on the parabola and y
3
 is the ordinate of the point of intersection of tangents 
at P and Q, then 
(a)  y
1
, y
2
, y
3
 are in A.P.               (b)  y
1
, y
3
, y
2
 are in A.P.               (c)  y
1
, y
2
, y
3
 are in G.P.            (d)  y
1
, y
3
, y
2
 are in 
G.P. 
18. The locus of the middle points of the focal chord of the parabola y
2
 = 4ax is 
(a)  y
2
 = 2a (x – a)  (b)  y
2
 = a (x – a)   (c)  y
2
 = 4a (x – a)  (d)  none of these 
19. The length of a local chord of the parabola y
2
 = 4ax making an angle ? with the axis of the parabola is 
(a)  a cosec
2
 ?  (b)  4a sec
2
 ?  (c)  4a cosec
2
 ?  (d)  None of these 
20. If the tangents at P and Q on a parabola meet in T, then SP, ST and SQ are in 
(a)  G.P.                      (b)  A.P.                     (c)  H.P.               (d)  None of these 
21. The point on the curve y
2
 = ax, the tangent at which makes an angle of 45° with x–axis will be given by 
 
 
22. If 2x + y + k = 0 is a normal to the parabola y
2
 = – 8x, then the value of k is 
(a)  – 16                                         (b)  – 8                                         (c)  24                                         (d)  – 24 
23. The equation of the ellipse having vertices at ( ± 5, 0), foci at ( ± 4, 0) is 
(a)  9x
2
 – 25y
2
 – 225 = 0     (b)  9x
2
 + 25y
2
 + 225 = 0       (c)  9x
2
 + 25y
2
 = 225  (d)  None of these 
24. The equation of the locus of all points the sum of whose distances from (3, 0) and (9, 0) is 12 is 
(a)  3x
2
 + 4y
2
 – 36x = 0     (b)  3x
2
 – 4y
2
 – 36x = 0               (c)  3x
2
 + 4y
2 
+ 36x = 0            (d)  3x
2
 + 36y
2
 – 4y = 0 
25. The eccentricity of the ellipse if its latus rectum is equal to one–half of its minor axis is 
 
 
26. The latus rectum of the ellipse 5x
2
 + 9y
2
 = 45 is 
(a)  10/3   (b)  5/3   (c)  2 v5 / 3   (d)  v5 / 3 
27. The length of the latus return of an ellipse is 1/3 of the major axis. Its eccentricity is 
 
 
28.  
 
 
29. The normals at three points P, Q, R of the parabola y
2
 = 4ax meet in (h, k). The centroid of triangle PQR lies on 
(a)  y = 0   (b)  x = 0   (c)  x = – a   (d)  y = a 
30. The equation of the tangent at the vertex of the parabola x
2
 + 4x + 2y = 0 is 
GIITJEE  (GURUS FOR IITJEE)  SCO 382, Sector 37–D, Chandigarh, Ph. 0172–2628810, 2628811, 2628820                         P  –  2 
 
The distances from the foci of P (x
1
, )
( ) ( ) ( ) ( )
y on the ellipse
x
y
are
a y b y c x d None of these
1
2
2
1 1 1
9 25
1
4
5
4
5
4
5
5
4
5
+ =
± ± ±
(a)  x = – 2   (b)  y = 2   (c)  x = 2   (d)  y = – 2 
 
 
 
 
 
 
 
 
 
 
 
ANSWERS  TO  CONIC  SECTION  (TEST  R1  :  + 
1)          3
rd
 Jan. 08 
1     (b) 11   (c) 21   (a) 
2    (b) 12   (a) 22   (c) 
3     (b) 13   (a) 23   (c) 
4     (a) 14   (b) 24   (a) 
5     (b) 15   (b) 25   (b) 
6     (a) 16   (a) 26   (a) 
7    (a) 17   (b) 27    (a) 
8     (b) 18   (a) 28    (a) 
9     (a) 19   (c) 29   (a) 
10   (a) 20   (a) 30   (b) 
 
GIITJEE  (GURUS FOR IITJEE)  SCO 382, Sector 37–D, Chandigarh, Ph. 0172–2628810, 2628811, 2628820                         P  –  3 
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