If the radius of the circle $x^2 + y^2 - 18x + 12y + k = 0$ is $11$,then $k = $

$A$ line is drawn through a fixed point $P(\alpha, \beta)$ to cut the circle $x^2 + y^2 = r^2$ at $A$ and $B$. Then $PA \cdot PB$ is equal to

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two points such that their abscissae $x_1$ and $x_2$ are the roots of the equation $x^2 + 2x - 3 = 0$,while the ordinates $y_1$ and $y_2$ are the roots of the equation $y^2 + 4y - 12 = 0$. The centre of the circle with $PQ$ as diameter is

The equation of the circle,the end-points of whose diameter are the centres of the circles $x^{2}+y^{2}-2x+3y-3=0$ and $x^{2}+y^{2}+6x-12y-5=0$ is

The equation of a circle of radius $5$ units touching another circle $x^2+y^2-2x-4y-20=0$ at $(5,5)$ is

What is the length of the diameter of the circle $x^2 + y^2 - 4x - 6y + 4 = 0$?