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

For the circle $x^2+y^2-9=0$,find the equation of the chord having $(1,2)$ as its mid-point.

The number of integers in the range of $m$ for which the point $(m, 1)$ lies in the smaller region enclosed by the circle $x^2 + y^2 - 3x + 1 = 0$ and the line $2x - y = 2$ is

Find the equation of the director circle of the circle $x^2 + y^2 = 8$.

Find the equation of a circle that touches the $Y$-axis at a distance of $4$ units from the origin and cuts an intercept of $6$ units on the $X$-axis.

In a $\triangle ABC$,a point $D$ is chosen on $BC$ such that $BD:DC = 2:5$. Let $P$ be a point on the circumcircle of $\triangle ABC$ such that $\angle PDB = \angle BAC$. Then $PD:PC$ is: