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

If one of the diameters of the circle given by the equation $x^{2}+y^{2}+4x+6y-12=0$ is a chord of a circle $S$ whose centre is $(2,-3)$, then the radius of $S$ is:

Let $OA$ be a radius of a circle with center $O$ and radius $d$. Let $B$ be a point on the circle such that $\angle AOB = \theta$ $(< \frac{\pi}{2})$. Let $D$ be a point on $OA$ such that $BD \perp OA$. Let $E$ be the mid-point of $BD$ and $F$ be a point on the arc $AB$ such that $EF \parallel OA$. Then,the ratio of the length of the arc $AF$ to the length of the arc $AB$ is

The number of circles that touch all the three lines $x+y-1=0$,$x-y-1=0$,and $y+1=0$ is

If the straight line $y=mx$ lies outside of the circle $x^2+y^2-20y+90=0$, then the value of $m$ will satisfy

$A$ circle with radius $12$ lies in the first quadrant and touches both the axes. Another circle has its centre at $(8, 9)$ and radius $7$. Which of the following statements is true?