The value of k so that ${x^2} + {y^2} + kx + 4y + 2 = 0$ and $2({x^2} + {y^2}) - 4x - 3y + k = 0$ cut orthogonally is

Number of common tangents to the circles
$x^2 + y^2 -2x + 4y -4 = 0$ and
$x^2 + y^2 -8x -4y + 16 = 0 $ is-

A circle ${C_1}$ of radius $2$ touches both $x$ - axis and $y$ - axis. Another circle ${C_2}$ whose radius is greater than $2$ touches circle ${C_1}$ and both the axes. Then the radius of circle ${C_2}$ is

The circle passing through point of intersection of the circle $S = 0$ and the line $P = 0$ is

Let $Z$ be the set of all integers,

$\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+\mathrm{y}^{2} \leq 4\right\}$

$\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}: \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\} \text { and }$

$\mathrm{C}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+(\mathrm{y}-2)^{2} \leq 4\right\}$

If the total number of relation from $\mathrm{A} \cap \mathrm{B}$ to $\mathrm{A} \cap \mathrm{C}$ is $2^{\mathrm{p}}$, then the value of $\mathrm{p}$ is :

Let $C_i \equiv  x^2 + y^2 = i^2 (i = 1,2,3)$ are three circles. If there are $4i$ points on circumference of circle $C_i$. If no three of all the points on three circles are collinear then number of triangles which can be formed using these points whose circumcentre does not lie on origin, is-