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

Match each entry in List-I to the correct entry in List-II and choose the correct option.

1. (P) → (3), (Q) → (4), (R) → (1), (S) → (2)
2. (P) → (3), (Q) → (2), (R) → (1), (S) → (5)
3. (P) → (3), (Q) → (2), (R) → (4), (S) → (5)
4. (P) → (4), (Q) → (1), (R) → (2), (S) → (3)

Consider the ellipse E given by x218+ y212=1. Let H be the hyperbola whose eccentricity is the reciprocal of the eccentricity of E and whose foci are the same as that of E. Let P and Q be the points of intersection of H and the parabola 5 y=x2 in the first quadrant. Let d be the distance between P and Q. If a and b are the integers such that d2=a+b 5, then the value of a-b is .