$A$ tangent to a circle intersects the circle in $\ldots \ldots \ldots .$

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 $cm$).

The chord of $\odot(O, 34)$ touches $\odot(O, 16)$. The length of the chord is .........

Two concentric circles having radii $5$ and $13$ are given. The chord of the circle with larger radius touches the circle with smaller radius. Then the length of the chord is $\ldots \ldots \ldots \ldots.$

If $d_{1}$ and $d_{2}$ $(d_{2} > d_{1})$ are the diameters of two concentric circles and $c$ is the length of a chord of the outer circle which is tangent to the inner circle,prove that $d_{2}^{2} = c^{2} + d_{1}^{2}$.

$A$ line passing through the centre of the circle $O$ intersects the tangent to the circle at $Q$. $P$ is the point of contact. If the radius of the circle is $9$ and $PQ = 40$,then find $OQ$.