In the figure,if $PA$ and $PB$ are tangents to the circle with center $O$ such that $\angle APB = 50^{\circ}$,then $\angle OAB$ is equal to: (in $^{\circ}$)

$\overline{PA}$ and $\overline{PB}$ are the tangents to $\odot(O, r)$ drawn from a point $P$ outside a circle. If $m \angle APB = 70^\circ$,then $m \angle POB = \dots$ (in $^\circ$)

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

Write True or False and give reasons for your answer.
$A$ pair of tangents can be constructed from a point $P$ to a circle of radius $3.5 \, cm$ situated at a distance of $3 \, cm$ from the centre.

$P$ is a point in the exterior of $\odot(O, r)$ and the tangents from $P$ to the circle touch the circle at $X$ and $Y$. Find $r$,if $OP = 30$ and $PY = 24$.

Write 'True' or 'False' and give reasons for your answer.
If the angle between two tangents drawn from a point $P$ to a circle of radius $a$ and center $O$ is $90^{\circ}$,then $OP = a\sqrt{2}$.