If $m$ is the slope and $P(8, \beta)$ is the midpoint of a chord of contact of the circle $x^2+y^2=125$,then the number of values of $\beta$ such that $\beta$ and $m$ are integers is

If the line $4x + 4y - 11 = 0$ intersects the circle $x^2 + y^2 - 4x - 6y + 4 = 0$ at $A$ and $B$,then the point of intersection of the tangents drawn at $A$ and $B$ is

If the common chord of the circles $x^2+y^2+4y=0$ and $x^2+y^2-4x-5=0$ is the diameter of the circle $S=0$,then the abscissa of the centre of the circle $S=0$ is

Find the condition that the pair of lines joining the origin to the points of intersection of the line $y = mx + c$ and the circle $x^{2} + y^{2} = a^{2}$ are at right angles to each other.

The circle $S \equiv x^2+y^2-2x-4y+1=0$ cuts the $y$-axis at $A, B$ $(OA > OB)$. If the radical axis of $S=0$ and $S^{\prime} \equiv x^2+y^2-4x-2y+4=0$ cuts the $y$-axis at $C$,then the ratio in which $C$ divides $AB$ is:

If $A$ and $B$ are the points of contact of the tangents drawn from the point $P(-3, 1)$ to the circle $x^2+y^2-4x+2y-4=0$,then the equation of the circumcircle of the triangle $PAB$ is