Conic Sections

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 .