Conic Sections

If the equation of the hyperbola with foci (4, 2) and (8, 2) is 3x2 - y2 - x + y + = 0, then + + is equal to .

If S and S′ are the foci of the ellipse x218 + y29 = 1 and P be a point on the ellipse, then (SP S'P) + (SP S'P) is equal to :

1. 3(6 + 2)
2. 3(1 + 2)
3. 27
4. 9

Let the focal chord PQ of the parabola y2 = 4x make an angle of 60 with the positive x axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y-axis at the point (0, ), then 5 2 is equal to:

1. 15
2. 25
3. 20
4. 30

Let one focus of the hyperbola H : x2a2 - y2b2 = 1 be at ( 10, 0) and the corresponding directrix be x = 9 10. If e and l respectively are the eccentricity and the length of the latus rectum of H, then 9(e2 + l) is equal to :

1. 12
2. 14
3. 15
4. 16

The absolute difference between the squares of the radii of the two circles passing through the point (-9, 4) and touching the lines x + y = 3 and x - y = 3, is equal to .

Let the point P of the focal chord PQ of the parabola y2 = 16x be (1, -4). If the focus of the parabola divides the chord PQ in the ratio m : n, (m, n) = 1, then m2 + n2 is equal to :

1. 17
2. 37
3. 10
4. 26

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

1. 3 19
2. 316
3. 4 17
4. 57

Let the ellipse E1 : x2a2 + y2b2 = 1, a > b and E2 : x2A2 + y2B2 = 1, A < B have same eccentricity 1 3. Let the product of their lengths of latus rectums be 32 3 and the distance between the foci of E1 be 4. If E1 and E2 meet at A, B, C and D, then the area of the quadrilateral ABCD equals :

1. 24 65
2. 18 65
3. 6 6
4. 12 65

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)2 is equal to :

1. 384
2. 392
3. 96
4. 192

Let the line x + y = 1 meet the circle x2 + y2 = 4 at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

1. 14
2. 3 7
3. 2 14
4. 5 7