Conic Sections

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

Let H : -x2a2 + y2b2 = 1 be the hyperbola, whose eccentricity is 3 and the length of the latus rectum is 4 3. Suppose the point ( , 6), > 0 lies on H. If is the product of the focal distances of the point ( , 6), then 2 + is equal to

1. 170
2. 171
3. 169
4. 172

Let S be the focus of the hyperbola x23 - y25 = 1, on the positive x-axis. Let C be the circle with its centre at A( 6, 5) and passing through the point S. If O is the origin and SAB is a diameter of C, then the square of the area of the triangle OSB is equal to .

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are m and n, respectively, then m + n2 is equal to

1. 408
2. 414
3. 312
4. 396

Let C be the circle of minimum area touching the parabola y = 6 - x2 and the lines y = 3|x|. Then, which one of the following points lies on the circle C ?

1. (1, 2)
2. (2, 2)
3. (1, 1)
4. (2, 4)

Let L1, L2 be the lines passing through the point P(0, 1) and touching the parabola 9x2 + 12x + 18y - 14 = 0. Let Q and R be the points on the lines L1 and L2 such that the PQR is an isosceles triangle with base QR. If the slopes of the lines QR are m1 and m2, then 16 (m12 + m22 ) is equal to .

The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and x = 4 3, respectively. Let the line y - 3x + 3 = 0 touch this hyperbola at (x0, y0). If m is the product of the focal distances of the point (x0, y0), then 4e2 + m is equal to .