If two circles touch each other externally,then $\ldots \ldots \ldots \ldots$ common tangents can be drawn to them.

$\overrightarrow{ PA }$ and $\overrightarrow{ PB }$ are tangents to $\odot( O , 5) .$ If $OP = 13,$ then $PB = \ldots \ldots \ldots \ldots$

In a right triangle $ABC$ in which $\angle B = 90^{\circ}$,a circle is drawn with $AB$ as diameter,intersecting the hypotenuse $AC$ at $P$. Prove that the tangent to the circle at $P$ bisects $BC$.

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

In $\Delta PQR$,$\angle Q$ is a right angle. If $PQ = 8$ and $QR = 15$,then the radius of a circle touching all the three sides of $\Delta PQR$ is $\ldots \ldots.$

Two concentric circles having radii $41$ and $40$ are given. The chord of the circle with larger radius touches the circle with smaller radius. Find the length of the chord.