Conic Sections

Consider a hyperbola H having centre at the origin and foci on the x-axis. Let C1 be the circle touching the hyperbola H and having the centre at the origin. Let C2 be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of C1 and C2 are 36 and 4 , respectively, then the length (in units) of latus rectum of H is

1. 283
2. 113
3. 143
4. 103

Let PQ be a chord of the parabola y2 = 12x and the midpoint of PQ be at (4, 1). Then, which of the following point lies on the line passing through the points P and Q ?

1. (3, -3)
2. ( 12, -20 )
3. (2, -9)
4. ( 32, -16 )

Let C be a circle with radius 10 units and centre at the origin. Let the line x + y = 2 intersect the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope -1. Then, a distance (in units) between the chord PQ and the chord MN is

1. 3 - 2
2. 2 - 3
3. 2 - 1
4. 2 + 1

For 0< < /2, if the eccentricity of the hyperbola x2-y2 cosec2 =5 is 7 times eccentricity of the ellipse x2 cosec2 +y2=5, then the value of is :

1. 6
2. 5 12
3. 3
4. 4

Let x2a2+ y2b2=1, a>b be an ellipse, whose eccentricity is 1 2 and the length of the latus rectum is 14. Then the square of the eccentricity of x2a2- y2b2=1 is :

1. 3
2. 72
3. 32
4. 52

Let C: x2+y2=4 and C': x2+y2-4 x+9=0 be two circles. If the set of all values of so that the circles C and C' intersect at two distinct points, is R-[a, b], then the point (8a+12, 16b-20) lies on the curve :

1. x2+2y2-5x+6y=3
2. 5x2-y=-11
3. x2-4y2=7
4. 6x2+y2=42

Let the line L: 2x+y= pass through the point of the intersection P (in the first quadrant) of the circle x2+y2=3 and the parabola x2=2y. Let the line L touch two circles C1 and C2 of equal radius 2 3. If the centres Q1 and Q2 of the circles C1 and C2 lie on the y-axis, then the square of the area of the triangle PQ1Q2 is equal to .

Let P be a point on the ellipse x29+ y24=1. Let the line passing through P and parallel to y-axis meet the circle x2+y2=9 at point Q such that P and Q are on the same side of the x-axis. Then, the eccentricity of the locus of the point R on PQ such that PR: RQ=4: 3 as P moves on the ellipse, is :

1. 1321
2. 13923
3. 137
4. 1119

Let the locus of the midpoints of the chords of the circle x2+(y-1)2=1 drawn from the origin intersect the line x+y=1 at P and Q. Then, the length of PQ is :

1. 12
2. 1
3. 1 2
4. 2

If one of the diameters of the circle x2+y2-10x+4y+13=0 is a chord of another circle C, whose center is the point of intersection of the lines 2x+3y=12 and 3x-2y=5, then the radius of the circle C is :

1. 4
2. 3 2
3. 6
4. 20