Let $C_1$ be the circle of radius $1$ with center at the origin. Let $C_2$ be the circle of radius $\mathrm{I}$ with center at the point $A=(4,1)$, where $1<\mathrm{r}<3$. Two distinct common tangents $P Q$ and $S T$ of $C_1$ and $C_2$ are drawn. The tangent $P Q$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $S T$ touches $C_1$ at $S$ and $C_2$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B=\sqrt{5}$, then the value of $r^2$ is

Suppose we have two circles of radius 2 each in the plane such that the distance between their centers is $2 \sqrt{3}$. The area of the region common to both circles lies between

The equation of the circle which passes through the origin, has its centre on the line $x + y = 4$ and cuts the circle ${x^2} + {y^2} - 4x + 2y + 4 = 0$ orthogonally, is

A circle $\mathrm{C}$ touches the line $\mathrm{x}=2 \mathrm{y}$ at the point $(2,1)$ and intersects the circle $C_{1}: x^{2}+y^{2}+2 y-5=0$ at two points $\mathrm{P}$ and $\mathrm{Q}$ such that $\mathrm{PQ}$ is a diameter of $\mathrm{C}_{1}$. Then the diameter of $\mathrm{C}$ is :

For the given circles ${x^2} + {y^2} - 6x - 2y + 1 = 0$ and ${x^2} + {y^2} + 2x - 8y + 13 = 0$, which of the following is true

The two circles ${x^2} + {y^2} - 2x + 6y + 6 = 0$ and ${x^2} + {y^2} - 5x + 6y + 15 = 0$