Conic Sections

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

Let A, B and C be three points on the parabola y2 = 6x and let the line segment AB meet the line L through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and B on L. Then ( AM BNCD )2 is equal to .

Consider the circle C : x2 + y2 = 4 and the parabola P : y2 = 8x. If the set of all values of , for which three chords of the circle C on three distinct lines passing through the point ( , 0) are bisected by the parabola P is the interval (p, q), then (2q - p)2 is equal to .

Let the circles C1 : (x - )2 + (y - )2 = r12 and C2 : (x - 8)2 + (y - 152 )2 = r22 touch each other externally at the point (6, 6). If the point (6, 6) divides the line segment joining the centres of the circles C1 and C2 internally in the ratio 2 : 1, then ( + ) + 4 (r12 + r22 ) equals

1. 130
2. 110
3. 145
4. 125