$\odot(O, 17)$ and $\odot(O, 15)$ are concentric circles. The chord $\overline{AB}$ of $\odot(O, 17)$ touches $\odot(O, 15)$. Then $AB = \ldots$

State 'True' or 'False' and give reasons for your answer.
If the angle between two tangents drawn from a point $P$ to a circle of radius $a$ and center $O$ is $60^{\circ}$,then $OP = a\sqrt{3}$.

In $\Delta ABC$,$\angle B = 90^{\circ}$. The radius of the incircle touching all three sides of the triangle is $\ldots \ldots \ldots \ldots$

Write 'True' or 'False' and give reasons for your answer.
If a chord $AB$ subtends an angle of $60^{\circ}$ at the centre of a circle,then the angle between the tangents at $A$ and $B$ is also $60^{\circ}$.

Point $P$ lies in the exterior of $\odot(O, r)$. $\overleftrightarrow{PQ}$ touches the circle at $Q$. If $PO = 26$ and $PQ = 10$,then find the diameter of the circle.

In the figure,$\stackrel{\leftrightarrow}{ AB }$,$\stackrel{\leftrightarrow}{ AC }$ and $\stackrel{\leftrightarrow}{ PQ }$ are tangents to $\odot( O , r)$. If the perimeter of $\Delta APQ$ is $16$,then $AB = \ldots$