Consider a family of circles which are passing through the point $(- 1, 1)$ and are tangent to $x-$ axis. If $(h, k)$ are the coordinate of the centre of the circles, then the set of values of $k$ is given by the interval

The circles ${x^2} + {y^2} - 10x + 16 = 0$ and ${x^2} + {y^2} = {r^2}$ intersect each other in two distinct points, if

Let a circle $C_1 \equiv  x^2 + y^2 - 4x + 6y + 1 = 0$ and circle $C_2$ is such that it's centre is image of centre of $C_1$ about $x-$axis and radius of $C_2$ is equal to radius of $C_1$, then area of $C_1$ which is not common with $C_2$ is -

The centre of the smallest circle touching the circles $x^2 + y^2- 2y - 3 = 0$ and $x^2+ y^2 - 8x - 18y + 93 = 0$ is :

Two given circles ${x^2} + {y^2} + ax + by + c = 0$ and ${x^2} + {y^2} + dx + ey + f = 0$ will intersect each other orthogonally, only when

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:

($1$) 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 ($2$)