Conic Sections

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 circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x+(k-1)y+3=0 and 2x+k2y-4=0. If the line x-y+2=0 intersects the circle at the points A and B, then (AB)2 is equal to :

1. 10
2. 27
3. 18
4. 34

Let O be the origin, and P and Q be two points on the rectangular hyperbola xy=12 such that the midpoint of the line segment PQ is ( 12, - 12 ). Then the area of the triangle OPQ equals :

1. 32
2. 52
3. 72
4. 92

Let the parabola y=x2+px+q passing through the point (1, -1) be such that the distance between its vertex and the x-axis is minimum. Then the value of p2+q2 is :

1. 2
2. 4
3. 5
4. 8

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