Conic Sections

Let a line perpendicular to the line 2x - y = 10 touch the parabola y2 = 4(x - 9) at the point P. The distance of the point P from the centre of the circle x2 + y2 - 14x - 8y + 56 = 0 is .

Let the length of the focal chord PQ of the parabola y2=12x be 15 units. If the distance of PQ from the origin is p, then 10p2 is equal to .

Let the line L: 2x+y= pass through the point of the intersection P (in the first quadrant) of the circle x2+y2=3 and the parabola x2=2y. Let the line L touch two circles C1 and C2 of equal radius 2 3. If the centres Q1 and Q2 of the circles C1 and C2 lie on the y-axis, then the square of the area of the triangle PQ1Q2 is equal to .

Let the foci and length of the latus rectum of an ellipse x2a2+ y2b2=1, a>b be ( 5, 0) and 50, respectively. Then, the square of the eccentricity of the hyperbola x2b2- y2a2b2=1 equals

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 .

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 .

Consider a circle (x- )2+(y- )2=50, where , >0. If the circle touches the line y+x=0 at the point P, whose distance from the origin is 4 2, then ( + )2 is equal to .