Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3 : (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions: (i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$, (ii) $C_1$ and $C_2$ both lie inside $C_3$, and (iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$. Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$. List-I: (I) $2h + k$ (II) $\frac{\text{Length of } ZW}{\text{Length of } XY}$ (III) $\frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$ (IV) $\alpha$ List-II: (P) $6$ (Q) $\sqrt{6}$ (R) $\frac{5}{4}$ (S) $\frac{21}{5}$ (T) $2\sqrt{6}$ (U) $\frac{10}{3}$ Which of the following is the only INCORRECT combination? (A) (I), (P) (B) (IV), (U) (C) (III), (R) (D) (IV), (S)
Let the circles $C_1 : x^2 + y^2 = 9$ and $C_2 : (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3 : (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions:
(i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$,
(ii) $C_1$ and $C_2$ both lie inside $C_3$, and
(iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.
Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$.
List-I:
(I) $2h + k$
(II) $\frac{\text{Length of } ZW}{\text{Length of } XY}$
(III) $\frac{\text{Area of triangle } MZN}{\text{Area of triangle } ZMW}$
(IV) $\alpha$
List-II:
(P) $6$
(Q) $\sqrt{6}$
(R) $\frac{5}{4}$
(S) $\frac{21}{5}$
(T) $2\sqrt{6}$
(U) $\frac{10}{3}$
Which of the following is the only INCORRECT combination?
(A) (I), (P)
(B) (IV), (U)
(C) (III), (R)
(D) (IV), (S)