Conic Sections

Let P be a parabola with vertex (2,3) and directrix 2x+y=6. Let an ellipse E: x2a2+ y2b2=1, a>b, of eccentricity 1 2 pass through the focus of the parabola P. Then, the square of the length of the latus rectum of E, is

1. 51225
2. 65625
3. 3858
4. 3478

If the circles (x+1)2+(y+2)2=r2 and x2+y2-4x-4y+4=0 intersect at exactly two distinct points, then

1. 12<r<7
2. 3<r<7
3. 5<r<9
4. 0<r<7

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

1. 1 3
2. 2 5
3. 32
4. 53

Let the latus rectum of the hyperbola x29- y2b2=1 subtend an angle of 3 at the centre of the hyperbola. If b2 is equal to lm(1+ n), where l and m are co-prime numbers, then l2+m2+n2 is equal to .

Let P be a point on the hyperbola H: x29- y24=1, in the first quadrant such that the area of triangle formed by P and the two foci of H is 2 13. Then, the square of the distance of P from the origin is

1. 26
2. 22
3. 20
4. 18

Let A( , 0) and B(0, ) be the points on the line 5 x+7 y=50. Let the point P divide the line segment A B internally in the ratio 7: 3. Let 3 x-25=0 be a directrix of the ellipse E: x2a2+ y2b2=1 and the corresponding focus be S. If from S, the perpendicular on the x-axis passes through P, then the length of the latus rectum of E is equal to,

1. 253
2. 259
3. 325
4. 329

Consider two circles C1: x2+y2=25 and C2:(x- )2+y2=16, where (5,9). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of C1 and C2 be -1 ( 638 ). If the length of common chord of C1 and C2 is , then the value of ( )2 equals .

Equations of two diameters of a circle are 2x-3y=5 and 3x-4y=7. The line joining the points (- 227,-4 ) and (- 17,3 ) intersects the circle at only one point P( , ). Then, 17 - is equal to .

If the points of intersection of two distinct conics x2+y2=4b and x216+ y2b2=1 lie on the curve y2=3x2, then 3 3 times the area of the rectangle formed by the intersection points is .

Let P( , ) be a point on the parabola y2=4x. If P also lies on the chord of the parabola x2=8y whose mid point is (1, 54 ), then ( -28)( -8) is equal to .