A line drawn through the point $A(5,7)$ cut the circle $x^2+y^2-36=0$ at the points $P$ and $Q$. Then, $A P \cdot A Q=$

Let $P$ be any point on the circle $x^2+y^2-2 x-1=0$ and $C$ be its centre. Let $A B$ be the chord of contact of $P$ with respect to the circle $x^2+y^2-2 x=0$. Then, the locus of the circumcentre of the $\triangle C A B$ is

If a circle $C$ passing through $(4,0)$ touches the circle $x^2+y^2+4 x-6 y-12=0$ externally at the point $(1$, -1 ), then the radius of $C$ is

If the circles $C_1: x^2+y^2+2 x+4 y-20=0$, $C_2: x^2+y^2+6 x-8 y+9=0$ have $n$ common tangents and the length of the tangent drawn from the centre of similitude to the circle $C_2$ is $l$, then $\frac{l}{n^2}=$