Conic Sections

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 .

If the shortest distance of the parabola y2=4x from the centre of the circle x2+y2-4x-16y+64=0 is d, then d2 is equal to :

1. 16
2. 24
3. 20
4. 36

The length of the chord of the ellipse x225+ y216=1, whose mid point is (1, 25 ), is equal to :

1. 16915
2. 20095
3. 15415
4. 17415

Four distinct points (2k,3k),(1,0),(0,1) and (0,0) lie on a circle for k equal to :

1. 313
2. 213
3. 513
4. 113