If the measure of the angle between two radii of the circle is $48^{\circ}$,then the measure of the angle between the tangents drawn at the end points of the radii is $\ldots \ldots \ldots$ (in $^{\circ}$)

In $\Delta ABC$,$\angle B$ is a right angle. If $AB = 8$ and $BC = 15$,find the radius of the incircle of $\Delta ABC$.

$\overline{AB}$ is a chord of $\odot(O, 13)$ such that $AB = 24$. Tangents at $A$ and $B$ to the circle intersect at $P$. Find $PA$.

The tangent at a point $C$ of a circle and a diameter $AB$ when extended intersect at $P$. If $\angle PCA = 110^{\circ},$ find $\angle CBA$ [see $Fig.$]. (in $^{\circ}$)

In the figure,from an external point $P$,a tangent $PT$ and a line segment $PAB$ are drawn to a circle with centre $O$. $ON$ is perpendicular to the chord $AB$. Prove that:
$(i) \quad PA \cdot PB = PN^2 - AN^2$
$(ii) \quad PN^2 - AN^2 = OP^2 - OT^2$
$(iii) \quad PA \cdot PB = PT^2$

Two circles with centres $O$ and $O^{\prime}$ of radii $3\, cm$ and $4\, cm$,respectively,intersect at two points $P$ and $Q$ such that $OP$ and $O^{\prime}P$ are tangents to the two circles. Find the length of the common chord $PQ$ (in $cm$).