Conic Sections

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