Conic Sections

The focus of the parabola y2=4x+16 is the centre of the circle C of radius 5. If the values of , for which C passes through the point of intersection of the lines 3x-y=0 and x+ y=4, are 1 and 2, 1< 2, then 12 1+29 2 is equal to .

Let the foci of a hyperbola be (1,14) and (1,-12). If it passes through the point (1,6), then the length of its latus-rectum is :

1. 256
2. 1445
3. 2885
4. 245

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2,5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval ( , ), then 3 -2 is equal to :

1. 10
2. 12
3. 14
4. 15

Let the parabola y=x2+px-3, meet the coordinate axes at the points P, Q and R. If the circle C with centre at (-1,-1) passes through the points P, Q and R, then the area of PQR is :

1. 4
2. 6
3. 5
4. 7

Let P(4,4 3) be a point on the parabola y2=4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

1. 34 33
2. 343 38
3. 17 3
4. 263 38

Let E: x2a2+ y2b2=1, a>b and H: x2A2- y2B2=1. Let the distance between the foci of E and the foci of H be 2 3. If a-A=2, and the ratio of the eccentricities of E and H is 13, then the sum of the lengths of their latus rectums is equal to :

1. 10
2. 7
3. 9
4. 8

Let a circle passing through (2, 0) have its centre at the point (h, k). Let (xc, yc ) be the point of intersection of the lines 3x + 5y = 1 and (2 + c)x + 5c2y = 1. If h = c 1 xc and k = c 1 yc, then the equation of the circle is :

1. 5x2 + 5y2 - 4x - 2y - 12 = 0
2. 25x2 + 25y2 - 20x + 2y - 60 = 0
3. 25x2 + 25y2 - 2x + 2y - 60 = 0
4. 5x2 + 5y2 - 4x + 2y - 12 = 0

Let f(x) = x2 + 9, g(x) = xx - 9 and a = f g(10), b = g f(3). If e and l denote the eccentricity and the length of the latus rectum of the ellipse x2a + y2b = 1, then 8e2 + l2 is equal to.

1. 6
2. 12
3. 8
4. 16

Let the centre of a circle, passing through the points (0, 0), (1, 0) and touching the circle x2 + y2 = 9, be (h, k). Then for all possible values of the coordinates of the centre (h, k), 4 (h2 + k2 ) is equal to .

Let the foci of a hyperbola H coincide with the foci of the ellipse E : (x - 1)2100 + (y - 1)275 = 1 and the eccentricity of the hyperbola H be the reciprocal of the eccentricity of the ellipse E. If the length of the transverse axis of H is and the length of its conjugate axis is , then 3 2 + 2 2 is equal to

1. 225
2. 237
3. 242
4. 205