Conic Sections

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are m and n, respectively, then m + n2 is equal to

1. 408
2. 414
3. 312
4. 396

Let C be the circle of minimum area touching the parabola y = 6 - x2 and the lines y = 3|x|. Then, which one of the following points lies on the circle C ?

1. (1, 2)
2. (2, 2)
3. (1, 1)
4. (2, 4)

Let L1, L2 be the lines passing through the point P(0, 1) and touching the parabola 9x2 + 12x + 18y - 14 = 0. Let Q and R be the points on the lines L1 and L2 such that the PQR is an isosceles triangle with base QR. If the slopes of the lines QR are m1 and m2, then 16 (m12 + m22 ) is equal to .

The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and x = 4 3, respectively. Let the line y - 3x + 3 = 0 touch this hyperbola at (x0, y0). If m is the product of the focal distances of the point (x0, y0), then 4e2 + m is equal to .

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is :

1. 4 2
2. 4
3. 5
4. 2 2

Let the line 2x + 3y - k = 0, k > 0, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is x2 + y2 - 3x - 2y = 0 and the length of the latus rectum of the ellipse x2 + 9y2 = k2 is mn, where m and n are coprime, then 2m + n is equal to

1. 12
2. 13
3. 11
4. 10

Suppose AB is a focal chord of the parabola y2 = 12x of length l and slope m < 3. If the distance of the chord AB from the origin is d, then l d2 is equal to .

Let the circle C1 : x2 + y2 - 2(x + y) + 1 = 0 and C2 be a circle having centre at (-1, 0) and radius 2. If the line of the common chord of C1 and C2 intersects the y-axis at the point P, then the square of the distance of P from the centre of C1 is:

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

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 .