Conic Sections

Consider the circle C : x2+y2-6x-8y-11=0. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle x2+y2- x- y- =0, then + +2 is equal to .

Let the centre of the circle x2+y2+2gx+2fy+25=0 be in the first quadrant and lie on the line 2x-y=4. Let the area of an equilateral triangle inscribed in the circle be 27 3. Then the square of the length of the chord of the circle on the line x=1 is .

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 .

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 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 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 .

Let A be the point (3, 0) and circles with variable diameter AB touch the circle x2+y2=36 internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then 72e2 is equal to .

For some (0, 2 ), let the eccentricity and the length of the latus rectum of the hyperbola x2 - y2 2 = 8 be e1 and l1, respectively, and let the eccentricity and the length of the latus rectum of the ellipse x2 2 + y2 = 6 be e2 and l2, respectively. If e12 = e22 ( 2 + 1 ), then ( l1 l2e1 e2 ) 2 is equal to .

Let (h, k) lie on the circle C : x2 + y2 = 4 and the point (2h + 1, 3k + 2) lie on an ellipse with eccentricity e. Then the value of 5e2 is equal to .

Let r be the radius of the circle, which touches x-axis at point (a, 0), a < 0 and the parabola y2 = 9x at the point (4, 6). Then r is equal to .

Consider the hyperbola x2a2 - y2b2 = 1 having one of its foci at P(-3, 0). If the latus rectum through its other focus subtends a right angle at P and a2 b2 = 2 - , , N, then + is .

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be (-5, 0) and 5x + 9 = 0, respectively. If the product of the focal distances of a point ( , 2 5) on the hyperbola is p, then 4p is equal to .

Let C be the circle x2 + (y - 1)2 = 2, E1 and E2 be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line x + y = 3 touch the curves C, E1 and E2 at P(x1, y1), Q(x2, y2) and R(x3, y3) respectively. Given that P is the mid point of the line segment QR and PQ = 2 23, the value of 9(x1y1 + x2y2 + x3y3) is equal to .

Let the product of the focal distances of the point P(4, 2 3) on the hyperbola H: x2a2 - y2b2 = 1 be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then p2 + q2 is equal to

If the equation of the hyperbola with foci (4, 2) and (8, 2) is 3x2 - y2 - x + y + = 0, then + + is equal to .

The absolute difference between the squares of the radii of the two circles passing through the point (-9, 4) and touching the lines x + y = 3 and x - y = 3, is equal to .

Let E1 : x29 + y24 = 1 be an ellipse. Ellipses Ei's are constructed such that their centres and eccentricities are same as that of E1, and the length of minor axis of Ei is the length of major axis of Ei+1 (i 1). If Ai is the area of the ellipse Ei, then 5 ( i=1 Ai ), is equal to .

Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B, and a denotes the area of SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is .