Conic Sections

Let P be the point on the parabola y = x2 such that the slope of the tangent to the parabola at the point P is 4 . Let Q be the point in the first quadrant lying on the circle x2 + y2 = 2 such that the slope of the tangent to the circle at the point Q is -1 . Let R be the point in the first quadrant lying on the ellipse x2 + 4y2 = 8 such that the slope of the tangent to the ellipse at the point R is - 12 . Then the radius of the circle passing through the points P, Q and R is

1. 10
2. 5
3. 52
4. 2 5

Let T be the tangent to the parabola y2=16x at the point (64,32). Let L be the tangent to the same parabola at another point (x1, y1) on the parabola. If L and T are perpendicular to each other, then the distance between the point (x1, y1) and the focus of the parabola, is

1. 154
2. 4
3. 174
4. 5

Let S denote the locus of the point of intersection of the pair of lines 4x-3y=12 , 4 x+3 y=12, where varies over the set of non-zero real numbers. Let T be the tangent to S passing through the points (p,0) and (0,q), q>0, and parallel to the line 4x- 3 2y=0. Then the value of pq is

1. -6 2
2. -3 2
3. -9 2
4. -12 2

Consider the ellipse x29+ y24=1. Let S(p,q) be a point in the first quadrant such that p29+ q24>1. Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive x-coordinate and O be the center of the ellipse. If the area of the triangle ORT is 32, then which of the following options is correct?

1. q=2, p=3 3
2. q=2, p=4 3
3. q=1, p=5 3
4. q=1, p=6 3