Conic Sections

Let O be the vertex of the parabola y2=4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is :

1. 1
2. 2
3. 4
4. 8

Let x2f (a2+7a+3 )+ y2f(3a+15)=1 represent an ellipse with major axis along y-axis, where f is a strictly decreasing positive function on R. If the set of all possible values of a is R-[ , ], then 2+ 2 is equal to :

1. 28
2. 40
3. 61
4. 24

Consider the circle C : x2+y2-6x-8y-11=0. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle x2+y2- x- y- =0, then + +2 is equal to .

Let one root of the quadratic equation in x : (k2-15k+27 )x2+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of the parabola y2=6kx is equal to :

1. 4
2. 6
3. 8
4. 12

If the eccentricity e of the hyperbola x2a2- y2b2=1, passing through (6, 4 3 ), satisfies 15 (e2+1 )=34e, then the length of the latus rectum of the hyperbola x2b2- y22 (a2+1 )=1 is :

1. 10
2. 20
3. 25
4. 30

Let chord PQ of length 3 13 of the parabola y2=12x be such that the ordinates of points P and Q are in the ratio 1 : 2. If the chord PQ subtends an angle at the focus of the parabola, then is equal to :

1. 35
2. 45
3. 513
4. 1213

Let the centre of the circle x2+y2+2gx+2fy+25=0 be in the first quadrant and lie on the line 2x-y=4. Let the area of an equilateral triangle inscribed in the circle be 27 3. Then the square of the length of the chord of the circle on the line x=1 is .

Let C be a circle having centre in the first quadrant and touching the x-axis at a distance of 3 units from the origin. If the circle C has an intercept of length 6 3 on y-axis, then the length of the chord of the circle C on the line x-y=3 is :

1. 8
2. 6
3. 6 2
4. 8 2

The eccentricity of an ellipse E with centre at the origin O is 32 and its directrices are x= 4 63. Let H: x2a2- y2b2=1 be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of E. Then the distance between the foci of H is :

1. 4 2 7
2. 4 27
3. 4 7
4. 87

Let x=9 be a directrix of an ellipse E, whose centre is at the origin and eccentricity is 13. Let P( , 0), >0, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is :

1. 9y2=8x(1-x)
2. 3y2=4x(1-x)
3. 9y2=8x(x-1)
4. 3y2=4x(x-1)

Let the line x-y=4 intersect the circle C: (x-4)2+(y+3)2=9 at the points Q and R. If P( , ) is a point on C such that PQ=PR, then (6 +8 )2 is equal to .

Let a focus of the ellipse E : x2a2+ y2b2=1 be S(4, 0) and its eccentricity be 45. If the point P(3, ) lies on E and O is the origin, then the area of POS is equal to:

1. 125
2. 145
3. 245
4. 485

Let P be a moving point on the circle x2+y2-6x-8y+21=0. Then, the maximum distance of P from the vertex of the parabola x2+6x+y+13=0 is equal to:

1. 8
2. 10
3. 12
4. 9

Let the point P be the vertex of the parabola y=x2-6x+12. If a line passing through the point P intersects the circle x2+y2-2x-4y+3=0 at the points R and S, then the maximum value of (PR+PS)2 is :

1. 10
2. 20
3. 25
4. 5

Let the directrix of the parabola P : y2=8x, cut x-axis at the point A. Let B( , ), >1, be a point on P such that the slope of AB is 3/5. If BC is a focal chord of P, then six times the area of ABC is :

1. 80
2. 160
3. 174
4. 192

Let the eccentricity e of a hyperbola satisfy the equation 6e2-11e+3=0. If the foci of the hyperbola are (3, 5) and (3, -4), then the length of its latus rectum is :

1. 113
2. 173
3. 152
4. 172

Suppose that two chords, drawn from the point (1, 2) on the circle x2+y2+x-3y=0 are bisected by the y-axis. If the other ends of these chords are R and S, and the mid point of the line segment RS is ( , ), then 6( + ) is equal to :

1. 1
2. 3
3. 4
4. 6

Consider the parabola P : y2=4kx and the ellipse E : x2a2+ y2b2=1. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is e, then e2+2 2 is equal to .

Let P(3 , 2 ), 0, be a point on the ellipse x29+ y24=1, Q be a point on the circle x2+y2-14x-14y+82=0 and R be a point on the line x+y=5 such that the centroid of the triangle PQR is (2+ , 3+ 23 ). Then the sum of the ordinates of all possible points R is :

1. 6
2. 2
3. 4
4. 8

Let H : x2a2- y2b2=1 be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is 83. If the line x= intersects the hyperbola H at the points A and B such that the area of the triangle AOB is 4 15, where O is the origin, then 2 equals

1. 12
2. 16
3. 24
4. 25

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola y2=16x. Let the vertex B containing the right angle be (4, 8) and the locus of the centroid of ABC be a conic C0. Then three times the length of latus rectum of C0 is .

Let an ellipse x2a2+ y2b2=1, a<b, pass through the point (4, 3) and have eccentricity 53. Then the length of its latus rectum is :

1. 4 53
2. 2 5
3. 7 53
4. 8 53

Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line x+y=1 is 14, then the square of the radius of C is .

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x+(k-1)y+3=0 and 2x+k2y-4=0. If the line x-y+2=0 intersects the circle at the points A and B, then (AB)2 is equal to :

1. 10
2. 27
3. 18
4. 34

Let O be the origin, and P and Q be two points on the rectangular hyperbola xy=12 such that the midpoint of the line segment PQ is ( 12, - 12 ). Then the area of the triangle OPQ equals :

1. 32
2. 52
3. 72
4. 92

Let the parabola y=x2+px+q passing through the point (1, -1) be such that the distance between its vertex and the x-axis is minimum. Then the value of p2+q2 is :

1. 2
2. 4
3. 5
4. 8

Let A be the point (3, 0) and circles with variable diameter AB touch the circle x2+y2=36 internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then 72e2 is equal to .

Let y = x be the equation of a chord of the circle C1 (in the closed half-plane x 0) of diameter 10 passing through the origin. Let C2 be another circle described on the given chord as its diameter. If the equation of the chord of the circle C2, which passes through the point (2, 3) and is farthest from the center of C2, is x + ay + b = 0, then a − b is equal to

1. −6
2. 10
3. 6
4. −2

For some (0, 2 ), let the eccentricity and the length of the latus rectum of the hyperbola x2 - y2 2 = 8 be e1 and l1, respectively, and let the eccentricity and the length of the latus rectum of the ellipse x2 2 + y2 = 6 be e2 and l2, respectively. If e12 = e22 ( 2 + 1 ), then ( l1 l2e1 e2 ) 2 is equal to .

Let the circle x2 + y2 = 4 intersect x-axis at the points A(a, 0), a > 0 and B(b, 0). Let P(2 , 2 ), 0 < < 2 and Q(2 , 2 ) be two points such that ( - ) = 2. Then the point of intersection of AQ and BP lies on :

1. x2 + y2 - 4x - 4 = 0
2. x2 + y2 - 4x - 4y = 0
3. x2 + y2 - 4x - 4y - 4 = 0
4. x2 + y2 - 4y - 4 = 0