Let the circle $C_1:x^2+y^2=9$ and $C_2:{\left(x-3\right)}^2+{\left(y-4\right)}^2=16$ intersect at the point X and Y. Suppose that another circle $C_3:{\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ satisfies the following conditions
(i). Centre of $C_3$ is collinear with the center of
(ii). \displaystyle{C}_{{{1}}}&{C}_{{{2}}}\displaystyle{C}_{{{1}}}&{C}_{{{2}}} both lie inside $C_3$ and
(iii). $C_3$ touches $C_1$ at M and $C_2$ at N
Let hte line through X and Y intersect $C_3$ at Z and W and let a common tangent of $C_1$ & $C_3$ be a tangent to the parabola $x^2=8\alpha y$
There are some expressions given in the following lists
$\begin{array}{cc}\text{List I}& \text{List II}\\ \left(I\right)2h+k& \left(P\right)6\\ \left(II\right)\frac{\text{length of ZW}}{\text{length of XY}}& \left(Q\right)\sqrt{6}\\ \left(III\right)\frac{\text{Area of }\Delta MZN}{\text{Area of }\Delta ZMW}& \left(R\right)\frac{5}{4}\\ \left(IV\right)\alpha & \left(S\right)\frac{21}{5}\\ & \left(T\right)2\sqrt{6}\\ & \left(U\right)\frac{10}{3}\end{array}$
Q. Which of the following is the only correct combination?
(A) (I)-(S)
(B) (II)-(Q) ltBrgt (C) (I)-(U)
(D) (II)-(T)

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