The length of the transverse common tangent of the circles $x^2+y^2-2x+4y+4=0$ and $x^2+y^2+4x-2y+1=0$ is

The maximum distance of the point $P(10, 7)$ from the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ is:

$A$ line meets the circle $x^2+y^2-4x-4y-8=0$ at two points $A$ and $B$. If $P(2,-2)$ is a point on the circle such that $PA=PB=2$,then the equation of the line $AB$ is:

Which of the following statements are true and which are false? In each case,give a valid reason for your answer.
$q$: The centre of a circle bisects each chord of the circle.

Let $C$ be the circle $x^2+y^2=1$ in the $XY$-plane. For each $t \geq 0$,let $L_t$ be the line passing through $(0,1)$ and $(t, 0)$. Note that $L_t$ intersects $C$ in two points,one of which is $(0,1)$. Let $Q_t$ be the other point. As $t$ varies between $1$ and $1+\sqrt{2}$,the collection of points $Q_t$ sweeps out an arc on $C$. The angle subtended by this arc at $(0,0)$ is

The equation of the circle which touches the circle $S \equiv x^2+y^2-10x-4y+19=0$ at the point $(2,3)$ internally and has a radius equal to half of the radius of the circle $S=0$ is: