$O$ is the centre of a circle. From an external point $P$ a straight line is drawn to meet the circle at $B$ and $A$. Prove that, $PA\times PB=P{O}^2-r^2$, where $r$ is the radius of the circle.

A ladder is placed on a wall in such a way that the top of the ladder touches the wall at a height of $24 m$. If the bottom of the ladder touches the ground at a distance $7 m$ from the wall, then the length of the ladder is

The angular bisectors of $\angle A$ and $\angle B$ of a parallelogram $ABCD$ meet at $O$. If $AC=12 cm$ and $BD=16 cm$. Then $AB$ is

[Note: question is wrong in the book]

In $∆ABC$, if $AB=\left(2p-1\right) cm, AC=2\sqrt{2p} cm$ and $BC=\left(2p+1\right) cm$ then $\angle BAC$ is

The centre of a circle is $O$. $PQ$ is a chord of the circle and the midpoint of the chord is $A$. If $OA=4 \mathrm{cm}$, and $PQ=6 \mathrm{cm}$, then the radius of the circle is