Let $C$ be a circle passing through the points $A (2,-1)$ and $B (3,4)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle $(x-5)^{2}+(y-1)^{2}=\frac{13}{2}$, then $r^{2}$ is equal to

A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2+y^2=16$ and $x^2+y^2=1$. Then

$(A)$ radius of $S$ is $8$

$(B)$ radius of $S$ is $7$

$(C)$ centre of $S$ is $(-7,1)$

$(D)$ centre of $S$ is $(-8,1)$

If $d$ is the distance between the centres of two circles, ${r_1},{r_2}$ are their radii and $d = {r_1} + {r_2}$, then

Coordinates of the centre of the circle which bisects the circumferences of the circles

$x^2 + y^2 = 1 ; x^2 + y^2 + 2x - 3 = 0$ and $x^2 + y^2 + 2y - 3 = 0$ is

If one common tangent of the two circles $x^2 + y^2 = 4$ and ${x^2} + {\left( {y - 3} \right)^2} = \lambda ,\lambda  > 0$ passes through the point $\left( {\sqrt 3 ,1} \right)$, then possible value of  $\lambda$ 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 :