Conic Sections Practice Questions - DPP for JEE

 Page 1


PART-I (Single Correct MCQs)
1. If from any point P, tangents PT, PT' are drawn to two given circles
with centres A and B respectively; and if PN is the perpendicular from
P on their radical axis, then 
(a) PN. AB
(b) 2PN. AB
(c) 4PN. AB
(d) None of these
2. The parametric form of equation of the circle
x
2
 + y
2
 – 6x + 2y – 28 = 0 is
(a)
(b)
(c)
(d)
3. The distance between the foci of an ellipse is 10 and its latus rectum is
15. Its equation referred to its axes as axes of coordinates is
Page 2


PART-I (Single Correct MCQs)
1. If from any point P, tangents PT, PT' are drawn to two given circles
with centres A and B respectively; and if PN is the perpendicular from
P on their radical axis, then 
(a) PN. AB
(b) 2PN. AB
(c) 4PN. AB
(d) None of these
2. The parametric form of equation of the circle
x
2
 + y
2
 – 6x + 2y – 28 = 0 is
(a)
(b)
(c)
(d)
3. The distance between the foci of an ellipse is 10 and its latus rectum is
15. Its equation referred to its axes as axes of coordinates is
(a) 3x
2
 + 4y
2
 = 300
(b) 2x
2
 + y
2
 = 50
(c) 10x
2
 + 15y
2
 = 300
(d) None of these
4. Two points P and Q are taken on the line joining the points A(0, 0) and
B(3a, 0) such that AP = PQ = QB. Circles are drawn on AP, PQ and QB
as diameters. The locus of the point S, the sum of the squares of the
tangents from which to the three circles is equal to b
2
, is
(a)
(b)
(c)
(d)
5. A hyperbola having the transverse axis of length 2 sin ?, is confocal
with the ellipse 3x
2
 + 4y
2
 = 12. Then its equation is
(a) x
2
 cosec
2
 ? – y
2
 sec
2
 ? = 1
(b) x
2
 sec
2
 ? – y
2
 cosec
2
 ? = 1
(c) x
2
 sin
2
 ? – y
2
 cos
2
 ? = 1
(d) x
2
 cos
2
 ?  – y
2
 sin
2
 ? = 1
6. If three points E, F, G are taken on the parabola y
2
 = 4ax so that their
ordinates are in G.P., then the tangents at E and G intersect on the
(a) directrix
(b) axis
(c) ordinate of F
(d) tangent at  F
7. The point ([P + 1], [P]) (where [x] is the greatest integer less than or
equal to x), lying inside the region bounded by the circle x
2
 + y
2
 – 2x –
15 = 0 and x
2
 + y
2
 – 2x – 7 = 0, then
(a) P ? [–1, 0) ? [0, 1) ? [1, 2)
(b) P ? [–1, 2) – {0, 1}
(c) P ? (–1, 2)
Page 3


PART-I (Single Correct MCQs)
1. If from any point P, tangents PT, PT' are drawn to two given circles
with centres A and B respectively; and if PN is the perpendicular from
P on their radical axis, then 
(a) PN. AB
(b) 2PN. AB
(c) 4PN. AB
(d) None of these
2. The parametric form of equation of the circle
x
2
 + y
2
 – 6x + 2y – 28 = 0 is
(a)
(b)
(c)
(d)
3. The distance between the foci of an ellipse is 10 and its latus rectum is
15. Its equation referred to its axes as axes of coordinates is
(a) 3x
2
 + 4y
2
 = 300
(b) 2x
2
 + y
2
 = 50
(c) 10x
2
 + 15y
2
 = 300
(d) None of these
4. Two points P and Q are taken on the line joining the points A(0, 0) and
B(3a, 0) such that AP = PQ = QB. Circles are drawn on AP, PQ and QB
as diameters. The locus of the point S, the sum of the squares of the
tangents from which to the three circles is equal to b
2
, is
(a)
(b)
(c)
(d)
5. A hyperbola having the transverse axis of length 2 sin ?, is confocal
with the ellipse 3x
2
 + 4y
2
 = 12. Then its equation is
(a) x
2
 cosec
2
 ? – y
2
 sec
2
 ? = 1
(b) x
2
 sec
2
 ? – y
2
 cosec
2
 ? = 1
(c) x
2
 sin
2
 ? – y
2
 cos
2
 ? = 1
(d) x
2
 cos
2
 ?  – y
2
 sin
2
 ? = 1
6. If three points E, F, G are taken on the parabola y
2
 = 4ax so that their
ordinates are in G.P., then the tangents at E and G intersect on the
(a) directrix
(b) axis
(c) ordinate of F
(d) tangent at  F
7. The point ([P + 1], [P]) (where [x] is the greatest integer less than or
equal to x), lying inside the region bounded by the circle x
2
 + y
2
 – 2x –
15 = 0 and x
2
 + y
2
 – 2x – 7 = 0, then
(a) P ? [–1, 0) ? [0, 1) ? [1, 2)
(b) P ? [–1, 2) – {0, 1}
(c) P ? (–1, 2)
(d) None of these
8. Let z = 1 – t + i , where t is a real parameter. The locus of z
in the argand plane is
(a) an ellipse
(b) hyperbola
(c) a straight line
(d) None of thees
9. The conic represented by the equation  is
(a) ellipse
(b) hyperbola
(c) parabola
(d) None of these
10. The line 3x + 2y + 1 = 0  meets the hyperbola  in the
points P and Q. The coordinates of the point of intersection of the
tangents at P and Q are
(a)
(b)
(c)
(d) None of these
11. The lengths of the tangent drawn from any point on the circle 
to the two circles
5x
2
 + 5y
2
 – 24x + 32y + 75 = 0 and 5x
2
 + 5y
2
 – 48x + 64y + 300 = 0 are
in the ratio of
(a) 1 : 2
(b) 2 : 3
(c) 3 : 4
(d) None of these
12. The equation of the parabola whose focus is (0, 0) and the tangent at the
vertex is x – y + 1 = 0 is
(a)
Page 4


PART-I (Single Correct MCQs)
1. If from any point P, tangents PT, PT' are drawn to two given circles
with centres A and B respectively; and if PN is the perpendicular from
P on their radical axis, then 
(a) PN. AB
(b) 2PN. AB
(c) 4PN. AB
(d) None of these
2. The parametric form of equation of the circle
x
2
 + y
2
 – 6x + 2y – 28 = 0 is
(a)
(b)
(c)
(d)
3. The distance between the foci of an ellipse is 10 and its latus rectum is
15. Its equation referred to its axes as axes of coordinates is
(a) 3x
2
 + 4y
2
 = 300
(b) 2x
2
 + y
2
 = 50
(c) 10x
2
 + 15y
2
 = 300
(d) None of these
4. Two points P and Q are taken on the line joining the points A(0, 0) and
B(3a, 0) such that AP = PQ = QB. Circles are drawn on AP, PQ and QB
as diameters. The locus of the point S, the sum of the squares of the
tangents from which to the three circles is equal to b
2
, is
(a)
(b)
(c)
(d)
5. A hyperbola having the transverse axis of length 2 sin ?, is confocal
with the ellipse 3x
2
 + 4y
2
 = 12. Then its equation is
(a) x
2
 cosec
2
 ? – y
2
 sec
2
 ? = 1
(b) x
2
 sec
2
 ? – y
2
 cosec
2
 ? = 1
(c) x
2
 sin
2
 ? – y
2
 cos
2
 ? = 1
(d) x
2
 cos
2
 ?  – y
2
 sin
2
 ? = 1
6. If three points E, F, G are taken on the parabola y
2
 = 4ax so that their
ordinates are in G.P., then the tangents at E and G intersect on the
(a) directrix
(b) axis
(c) ordinate of F
(d) tangent at  F
7. The point ([P + 1], [P]) (where [x] is the greatest integer less than or
equal to x), lying inside the region bounded by the circle x
2
 + y
2
 – 2x –
15 = 0 and x
2
 + y
2
 – 2x – 7 = 0, then
(a) P ? [–1, 0) ? [0, 1) ? [1, 2)
(b) P ? [–1, 2) – {0, 1}
(c) P ? (–1, 2)
(d) None of these
8. Let z = 1 – t + i , where t is a real parameter. The locus of z
in the argand plane is
(a) an ellipse
(b) hyperbola
(c) a straight line
(d) None of thees
9. The conic represented by the equation  is
(a) ellipse
(b) hyperbola
(c) parabola
(d) None of these
10. The line 3x + 2y + 1 = 0  meets the hyperbola  in the
points P and Q. The coordinates of the point of intersection of the
tangents at P and Q are
(a)
(b)
(c)
(d) None of these
11. The lengths of the tangent drawn from any point on the circle 
to the two circles
5x
2
 + 5y
2
 – 24x + 32y + 75 = 0 and 5x
2
 + 5y
2
 – 48x + 64y + 300 = 0 are
in the ratio of
(a) 1 : 2
(b) 2 : 3
(c) 3 : 4
(d) None of these
12. The equation of the parabola whose focus is (0, 0) and the tangent at the
vertex is x – y + 1 = 0 is
(a)
(b)
(c)
(d)
13. The curve described parametrically by x = 2 – 3 sec t,y = 1 + 4 tan t
represents :
(a) An ellipse centred at (2, 1) and of eccentricity 
(b) A circle centred at (2, 1) and of radius 5 units
(c) A hyperbola centred at (2, 1) & of eccentricity 
(d) A hyperbola centred at (2, 1) & of eccentricity 
14. If a circle passes through the point (a, b) and cuts the circle 
orthogonally, then the locus of its centre is
(a)
(b)
(c)
(d)
15. If a variable point P on an ellipse of eccentricity e is joined to the foci
S
1
 and S
2
 then the incentre of the triangle PS
1
S
2
 lies on
(a) The major axis of the ellipse
(b) The circle with radius e
(c) Another ellipse of eccentricity 
(d) None of these
16. From the origin, chords are drawn to the circle
(x – 1)
2
 + y
2
 = 1, then equation of locus of middle points of these chords, is -
(a) x
2
 + y
2
 = 1
(b) x
2
 + y
2
 = x
Page 5


PART-I (Single Correct MCQs)
1. If from any point P, tangents PT, PT' are drawn to two given circles
with centres A and B respectively; and if PN is the perpendicular from
P on their radical axis, then 
(a) PN. AB
(b) 2PN. AB
(c) 4PN. AB
(d) None of these
2. The parametric form of equation of the circle
x
2
 + y
2
 – 6x + 2y – 28 = 0 is
(a)
(b)
(c)
(d)
3. The distance between the foci of an ellipse is 10 and its latus rectum is
15. Its equation referred to its axes as axes of coordinates is
(a) 3x
2
 + 4y
2
 = 300
(b) 2x
2
 + y
2
 = 50
(c) 10x
2
 + 15y
2
 = 300
(d) None of these
4. Two points P and Q are taken on the line joining the points A(0, 0) and
B(3a, 0) such that AP = PQ = QB. Circles are drawn on AP, PQ and QB
as diameters. The locus of the point S, the sum of the squares of the
tangents from which to the three circles is equal to b
2
, is
(a)
(b)
(c)
(d)
5. A hyperbola having the transverse axis of length 2 sin ?, is confocal
with the ellipse 3x
2
 + 4y
2
 = 12. Then its equation is
(a) x
2
 cosec
2
 ? – y
2
 sec
2
 ? = 1
(b) x
2
 sec
2
 ? – y
2
 cosec
2
 ? = 1
(c) x
2
 sin
2
 ? – y
2
 cos
2
 ? = 1
(d) x
2
 cos
2
 ?  – y
2
 sin
2
 ? = 1
6. If three points E, F, G are taken on the parabola y
2
 = 4ax so that their
ordinates are in G.P., then the tangents at E and G intersect on the
(a) directrix
(b) axis
(c) ordinate of F
(d) tangent at  F
7. The point ([P + 1], [P]) (where [x] is the greatest integer less than or
equal to x), lying inside the region bounded by the circle x
2
 + y
2
 – 2x –
15 = 0 and x
2
 + y
2
 – 2x – 7 = 0, then
(a) P ? [–1, 0) ? [0, 1) ? [1, 2)
(b) P ? [–1, 2) – {0, 1}
(c) P ? (–1, 2)
(d) None of these
8. Let z = 1 – t + i , where t is a real parameter. The locus of z
in the argand plane is
(a) an ellipse
(b) hyperbola
(c) a straight line
(d) None of thees
9. The conic represented by the equation  is
(a) ellipse
(b) hyperbola
(c) parabola
(d) None of these
10. The line 3x + 2y + 1 = 0  meets the hyperbola  in the
points P and Q. The coordinates of the point of intersection of the
tangents at P and Q are
(a)
(b)
(c)
(d) None of these
11. The lengths of the tangent drawn from any point on the circle 
to the two circles
5x
2
 + 5y
2
 – 24x + 32y + 75 = 0 and 5x
2
 + 5y
2
 – 48x + 64y + 300 = 0 are
in the ratio of
(a) 1 : 2
(b) 2 : 3
(c) 3 : 4
(d) None of these
12. The equation of the parabola whose focus is (0, 0) and the tangent at the
vertex is x – y + 1 = 0 is
(a)
(b)
(c)
(d)
13. The curve described parametrically by x = 2 – 3 sec t,y = 1 + 4 tan t
represents :
(a) An ellipse centred at (2, 1) and of eccentricity 
(b) A circle centred at (2, 1) and of radius 5 units
(c) A hyperbola centred at (2, 1) & of eccentricity 
(d) A hyperbola centred at (2, 1) & of eccentricity 
14. If a circle passes through the point (a, b) and cuts the circle 
orthogonally, then the locus of its centre is
(a)
(b)
(c)
(d)
15. If a variable point P on an ellipse of eccentricity e is joined to the foci
S
1
 and S
2
 then the incentre of the triangle PS
1
S
2
 lies on
(a) The major axis of the ellipse
(b) The circle with radius e
(c) Another ellipse of eccentricity 
(d) None of these
16. From the origin, chords are drawn to the circle
(x – 1)
2
 + y
2
 = 1, then equation of locus of middle points of these chords, is -
(a) x
2
 + y
2
 = 1
(b) x
2
 + y
2
 = x
(c) x
2
 + y
2
 = y
(d) None of these
17. The combined equation of the asymptotes of the hyperbola 2x
2
 + 5xy +
2y
2
 + 4x + 5y = 0 is –
(a) 2x
2
 + 5xy + 2y
2
 + 4x + 5y + 2 = 0
(b) 2x
2
 + 5xy + 2y
2
 + 4x + 5y – 2 = 0
(c) 2x
2
 + 5xy + 2y
2
 = 0
(d) None of these
18. The equation of one of the common tangents to the parabola y
2
 = 8x and
 is
(a) y = –x + 2
(b) y = x – 2
(c) y = x + 2
(d) None of these
19. If the axes of an ellipse coincides with the co-ordiante axes and it
passes through the point (4, –1) and touches the line x + 4y – 10 = 0
then the eq. is
(a)
(b)
(c)
(d) Both (a) and (b)
20. The equation of the image of circle
x
2
 + y
2 
+ 16x – 24y + 183 = 0 by the line  mirror
4x + 7y + 13 = 0 is
(a)
(b)
(c)