Answer the following by appropriately matching the lists based on the information given in the paragraph

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 $x^2=8 \alpha y$.

There are some expression given in the $List-I$ whose values are given in $List-II$ below:

$List-I$	$List-II$
$(I)$ $2 h + k$	$(P)$ $6$
$(II)$ $\frac{\text { Length of } ZW }{\text { Length of } XY }$	$(Q)$ $\sqrt{6}$
$(III)$ $\frac{\text { Area of triangle } MZN }{\text { Area of triangle ZMW }}$	$(R)$ $\frac{5}{4}$
$(IV)$ $\alpha$	$(S)$ $\frac{21}{5}$
	$(T)$ $2 \sqrt{6}$
	$(U)$ $\frac{10}{3}$

($1$) Which of the following is the only INCORRECT combination?

$(1) (IV), (S)$ $(2) (IV), (U)$ $(3) (III), (R)$ $(4) (I), (P)$

($2$) Which of the following is the only CORRECT combination?

$(1) (II), (T)$ $(2) (I), (S)$ $(3) (I), (U)$ $(4) (II), (Q)$

Give the answer or quetion ($1$) and ($2$)

• A

  $1,4$

• B

  $1,3$

• C

  $1,2$

• D

  $2,4$