Conic Sections · Question 61

Let e1 and e2 be the eccentricities of the ellipse x2b2 + y225 = 1 and the hyperbola x216 - y2b2 = 1, respectively. If b < 5 and e1 e2 = 1, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

1. 45
2. 35
3. 74
4. 32

Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If p + q = 126, then the eccentricity of the ellipse x216 + y2n = 1 is :

1. 1 2
2. 12
3. 74
4. 34

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be (-5, 0) and 5x + 9 = 0, respectively. If the product of the focal distances of a point ( , 2 5) on the hyperbola is p, then 4p is equal to .

The length of the latus-rectum of the ellipse, whose foci are (2, 5) and (2, -3) and eccentricity is 45, is

1. 503
2. 185
3. 65
4. 103

Let C be the circle x2 + (y - 1)2 = 2, E1 and E2 be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line x + y = 3 touch the curves C, E1 and E2 at P(x1, y1), Q(x2, y2) and R(x3, y3) respectively. Given that P is the mid point of the line segment QR and PQ = 2 23, the value of 9(x1y1 + x2y2 + x3y3) is equal to .

Let the sum of the focal distances of the point P(4, 3) on the hyperbola H: x2a2 - y2b2 = 1 be 8 53. If for H, the length of the latus rectum is l and the product of the focal distances of the point P is m, then 9l2 + 6m is equal to :

1. 187
2. 184
3. 186
4. 185

Let for two distinct values of p the lines y = x + p touch the ellipse E: x242 + y232 = 1 at the points A and B. Let the line y = x intersect E at the points C and D. Then the area of the quadrilateral ABCD is equal to :

1. 48
2. 20
3. 24
4. 36

The centre of a circle C is at the centre of the ellipse E: x2a2 + y2b2 = 1, a > b. Let C pass through the foci F1 and F2 of E such that the circle C and the ellipse E intersect at four points. Let P be one of these four points. If the area of the triangle PF1F2 is 30 and the length of the major axis of E is 17, then the distance between the foci of E is :

1. 12
2. 26
3. 13
4. 132

The axis of a parabola is the line y=x and its vertex and focus are in the first quadrant at distances 2 and 2 2 units from the origin, respectively. If the point (1, k) lies on the parabola, then a possible value of k is :

1. 8
2. 3
3. 9
4. 4

The radius of the smallest circle which touches the parabolas y = x2 + 2 and x = y2 + 2 is

1. 7 216
2. 7 28
3. 7 22
4. 7 24