Conic Sections

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

Let the length of the latus rectum of an ellipse x2a2 + y2b2 = 1, (a > b), be 30. If its eccentricity is the maximum value of the function f(t) = - 34 + 2t - t2, then (a2 + b2) is equal to

1. 276
2. 516
3. 256
4. 496

Let (h, k) lie on the circle C : x2 + y2 = 4 and the point (2h + 1, 3k + 2) lie on an ellipse with eccentricity e. Then the value of 5e2 is equal to .

Let the domain of the function f(x) = 3 5 7 (9x - x2 - 13 ) be the interval (m, n). Let the hyperbola x2a2 - y2b2 = 1 have eccentricity n3 and the length of the latus rectum 8m3. Then b2 - a2 is equal to :

1. 7
2. 9
3. 11
4. 5

Let the line y - x = 1 intersect the ellipse x22 + y21 = 1 at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is :

1. - -1 ( 14 )
2. 2 + -1 ( 14 )
3. 2 + 2 -1 ( 14 )
4. 2 - -1 ( 14 )

Let PQ be a chord of the hyperbola x24 - y2b2 = 1, perpendicular to the x-axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is 3, then the area of the triangle OPQ is

1. 2 3
2. 115
3. 8 35
4. 95

If the points of intersection of the ellipses x2 + 2y2 - 6x - 12y + 23 = 0 and 4x2 + 2y2 - 20x - 12y + 35 = 0 lie on a circle of radius r and centre (a, b), then the value of ab + 18r2 is :

1. 53
2. 52
3. 55
4. 51

An equilateral triangle OAB is inscribed in the parabola y2 = 4x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

1. 2(8 - 3 3)
2. 4(6 + 3)
3. 4(3 - 3)
4. 2(3 + 3)

If the line x + 2y = 1, where R, does not meet the hyperbola x2 - 9y2 = 9, then a possible value of is :

1. 0.6
2. 0.7
3. 0.8
4. 0.5