Conic Sections · Question 26

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 .

Let y = x be the equation of a chord of the circle C1 (in the closed half-plane x 0) of diameter 10 passing through the origin. Let C2 be another circle described on the given chord as its diameter. If the equation of the chord of the circle C2, which passes through the point (2, 3) and is farthest from the center of C2, is x + ay + b = 0, then a − b is equal to

1. −6
2. 10
3. 6
4. −2

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 the circle x2 + y2 = 4 intersect x-axis at the points A(a, 0), a > 0 and B(b, 0). Let P(2 , 2 ), 0 < < 2 and Q(2 , 2 ) be two points such that ( - ) = 2. Then the point of intersection of AQ and BP lies on :

1. x2 + y2 - 4x - 4 = 0
2. x2 + y2 - 4x - 4y = 0
3. x2 + y2 - 4x - 4y - 4 = 0
4. x2 + y2 - 4y - 4 = 0

An ellipse has its center at (1, -2), one focus at (3, -2) and one vertex at (5, -2). Then the length of its latus rectum is :

1. 6
2. 6 3
3. 16 3
4. 4 3

Let A be the focus of the parabola y2 = 8x. Let the line y = mx + c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is ( 73, 43 ), then (BC)2 is equal to :

1. 89
2. 80
3. 32
4. 41

Let the ellipse E : x2144 + y2169 = 1 and the hyperbola H : x216 - y2 2 = -1 have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H, then the value of 24(e + L) is :

1. 296
2. 126
3. 67
4. 148

Let a circle of radius 4 pass through the origin O, the points A(- 3a, 0) and B(0, - 2b), where a and b are real parameters and ab 0. Then the locus of the centroid of OAB is a circle of radius

1. 73
2. 113
3. 53
4. 83

Let each of the two ellipses E1: x2a2 + y2b2 = 1, (a > b) and E2: x2A2 + y2B2 = 1, (A < B) have eccentricity 45. Let the lengths of the latus recta of E1 and E2 be l1 and l2, respectively, such that 2l12 = 9l2. If the distance between the foci of E1 is 8, then the distance between the foci of E2 is

1. 965
2. 85
3. 165
4. 325

Let the image of parabola x2 = 4y, in the line x - y = 1 be (y + a)2 = b(x - c), a, b, c N. Then a + b + c is equal to

1. 12
2. 8
3. 6
4. 4