Conic Sections

Let H1: x2a2- y2b2=1 and H2:- x2A2+ y2B2=1 be two hyperbolas having length of latus rectums 15 2 and 12 5 respectively. Let their eccentricities be e1= 52 and e2 respectively. If the product of the lengths of their transverse axes is 100 10, then 25e22 is equal to .

If the line 3x-2y+12=0 intersects the parabola 4y=3x2 at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to

1. 2- -1 ( 32 )
2. -1 ( 97 )
3. -1 ( 119 )
4. -1 ( 45 )

Let the circle C touch the line x-y+1=0, have the centre on the positive x-axis, and cut off a chord of length 4 13 along the line -3x+2y=1. Let H be the hyperbola x2 2- y2 2=1, whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then 2 2+3 2 is equal to .

The length of the chord of the ellipse x24+ y22=1, whose mid-point is (1, 12 ), is :

1. 23 15
2. 13 15
3. 15
4. 53 15

Let the shortest distance from (a,0), a>0, to the parabola y2=4x be 4. Then the equation of the circle passing through the point (a,0) and the focus of the parabola, and having its centre on the axis of the parabola is :

1. x2+y2-8x+7=0
2. x2+y2-6x+5=0
3. x2+y2-4x+3=0
4. x2+y2-10x+9=0

The focus of the parabola y2=4x+16 is the centre of the circle C of radius 5. If the values of , for which C passes through the point of intersection of the lines 3x-y=0 and x+ y=4, are 1 and 2, 1< 2, then 12 1+29 2 is equal to .

Let the foci of a hyperbola be (1,14) and (1,-12). If it passes through the point (1,6), then the length of its latus-rectum is :

1. 256
2. 1445
3. 2885
4. 245

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2,5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval ( , ), then 3 -2 is equal to :

1. 10
2. 12
3. 14
4. 15

Let the parabola y=x2+px-3, meet the coordinate axes at the points P, Q and R. If the circle C with centre at (-1,-1) passes through the points P, Q and R, then the area of PQR is :

1. 4
2. 6
3. 5
4. 7

Let P(4,4 3) be a point on the parabola y2=4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

1. 34 33
2. 343 38
3. 17 3
4. 263 38