If a direct common tangent drawn to the circles $x^2+y^2-6x+4y+9=0$ and $x^2+y^2+2x-2y+1=0$ touches the circles at $A$ and $B$,then $AB=$

$A$ circle passing through the point $(1,0)$ makes an intercept of length $4$ units on the $X$-axis and an intercept of length $2\sqrt{11}$ units on the $Y$-axis. If the centre of the circle lies in the fourth quadrant,then the radius of the circle is

The area (in sq. units) of the region bounded by the circle $x^2+y^2=64$, positive $x$-axis and the line $y=\sqrt{3}x$ is

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

In a $\triangle ABC$,a point $D$ is chosen on $BC$ such that $BD:DC = 2:5$. Let $P$ be a point on the circumcircle of $\triangle ABC$ such that $\angle PDB = \angle BAC$. Then $PD:PC$ is:

If a circle with radius $2.5$ units passes through the points $(2, 3)$ and $(5, 7)$,then its centre is