Conic Sections

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :

1. r = 1
2. 2r2 - 4r + 1 = 0
3. 2r2 - 8r + 7 = 0
4. r2 - 8r + 8 = 0

Let the maximum and minimum values of ( 8x - x2 - 12 - 4 )2 + (x - 7)2, x R be M and m, respectively. Then M2 - m2 is equal to .

Let a line perpendicular to the line 2x - y = 10 touch the parabola y2 = 4(x - 9) at the point P. The distance of the point P from the centre of the circle x2 + y2 - 14x - 8y + 56 = 0 is .

A square is inscribed in the circle x2+y2-10x-6y+30=0. One side of this square is parallel to y=x+3. If (xi, yi ) are the vertices of the square, then (xi2+yi2 ) is equal to:

1. 152
2. 148
3. 156
4. 160

Let the length of the focal chord PQ of the parabola y2=12x be 15 units. If the distance of PQ from the origin is p, then 10p2 is equal to .

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