Two concentric circles of radii $a$ and $b$ ($a>b$) are given. Find the length of the chord of the larger circle which touches the smaller circle.

Write true or false and justify your answer in the following statement:

If a chord subtends an angle of $60°$ at the centre of a circle, then the angle between the tangents at A and B is also $60°$.

Write true or false and justify your answer in the following statement:

The angle between two tangents to a circle may be $0°$.

Write true or false and justify your answer in the following statement:

If angle between two tangents drawn from a point $P$ to a circle of radius $a$ and centre $O$ is $90°$, then $OP=\sqrt{2}a$.

In the adjoining figure, $PA$ and $PB$ are two tangents from an external point to a circle with centre $C$ and radius $4 \mathrm{cm}$. If $PA\perp PB$, then find the length of each tangent.

In the figure given below, $PQ$ is a tangent to a circle with centre $O$. If $\angle OAB=30°$, find $\angle ABP$ and $\angle AOB$.

In the figure given below, $O$ is the centre of a circle. $PQ$ is a chord and $PT$ is a tangent at $P$. If $\angle POQ=70°$, then find $\angle TPQ$.