In a triangle,two vertices are $(2, 3)$ and $(4, 0)$,and its circumcentre is $(2, z)$ for some real number $z$. The circumradius is

The points $A, B, C, D, E$ are marked on the circumference of a circle in clockwise direction such that $\angle ABC = 130^{\circ}$ and $\angle CDE = 110^{\circ}$. The measure of $\angle ACE$ in degrees is: (in $^{\circ}$)

Observe the following statements:
$I$. The circle $x^2+y^2-6x-4y-7=0$ touches the $y$-axis.
$II$. The circle $x^2+y^2+6x+4y-7=0$ touches the $x$-axis.
Which of the following is a correct statement?

$A$ 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:

$A$ line passing through the point $P(3, 11)$ intersects the circle $x^{2} + y^{2} = 9$ at points $A$ and $B$. Then $PA \cdot PB = . . . . .$

Let a circle $C$ of radius $1$ and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $C$ from the point $(5,5)$ is: