Conic Sections

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

Let the set of all values of r, for which the circles (x+1)2 + (y+4)2 = r2 and x2 + y2 - 4x - 2y - 4 = 0 intersect at two distinct points be the interval ( , ). Then is equal to

1. 21
2. 24
3. 20
4. 25

If the chord joining the points P1(x1, y1) and P2(x2, y2) on the parabola y2 = 12x subtends a right angle at the vertex of the parabola, then x1x2 - y1y2 is equal to

1. 280
2. 288
3. 292
4. 284

Let the locus of the mid-point of the chord through the origin O of the parabola y2 = 4x be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3 : 1, is :

1. 2x2 = 3y
2. 2y2 = 3x
3. 3y2 = 2x
4. 3x2 = 2y

Let S and S′ be the foci of the ellipse x225 + y29 = 1 and P( , ) be a point on the ellipse in the first quadrant. If (SP)2 + (S'P)2 - SP S'P = 37, then 2 + 2 is equal to :

1. 13
2. 15
3. 11
4. 17

Let P(10, 2 15) be a point on the hyperbola x2a2 - y2b2 = 1, whose foci are S and S'. If the length of its latus rectum is 8, then the square of the area of PSS' is equal to :

1. 4200
2. 1462
3. 900
4. 2700