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.

If the radii of two concentric circles are $4 \, cm$ and $5 \, cm$,then the length of each chord of the larger circle which is tangent to the smaller circle is (in $cm$):

$A$ circle with centre $O$ touches sides $\overline{AB}$,$\overline{BC}$,$\overline{CD}$ and $\overline{DA}$ of quadrilateral $ABCD$ at points $P, Q, R$ and $S$ respectively. Prove that $m \angle AOB + m \angle COD = 180^{\circ}$ and $m \angle AOD + m \angle BOC = 180^{\circ}$.

$PQ$ is a chord of length $8 \, cm$ of a circle of radius $5 \, cm$. The tangents at $P$ and $Q$ intersect at a point $T$ (see figure). Find the length $TP$.

Two concentric circles are of radii $5 \, cm$ and $3 \, cm$. Find the length of the chord of the larger circle which touches the smaller circle.

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 $^{\circ}$)