$P$ 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 a chord $AB$ subtends an angle of $60^{\circ}$ at the centre of a circle,then the angle between the tangents at $A$ and $B$ is also $60^{\circ}$.

$P$ lies in the exterior of $\odot(O, 8)$ such that $OP = 17$. Two tangents are drawn to the circle which touch the circle at $A$ and $B$. Find $AB$.

If the angle between two radii of a circle is $130^{\circ}$,the angle between the tangents at the ends of the radii is: (in $^{\circ}$)

$A$ is a point at a distance $13\, cm$ from the centre $O$ of a circle of radius $5 \,cm$. $AP$ and $AQ$ are the tangents to the circle at $P$ and $Q$. If a tangent $BC$ is drawn at a point $R$ lying on the minor arc $PQ$ to intersect $AP$ at $B$ and $AQ$ at $C$,find the perimeter of the $\triangle ABC$. (in $cm$)

State '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 $60^{\circ}$,then $OP = a\sqrt{3}$.