$P$ is in the exterior of $\odot (O, 9)$. $A$ tangent from $P$ touches the circle at $T$. If $PT = 40$,then $OP = \ldots$

If $\square ABCD$ is a cyclic quadrilateral and also a rectangle,and if $AB = 5$ and $BC = 12$,then $AC = \ldots$

$\overline{PA}$ is a tangent to $\odot(O, r)$ drawn from a point $P$ outside a circle. If $OP = 10$ and $AP = 8$,then the diameter of the circle is equal to $\ldots \ldots \ldots \ldots .$

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 $d_{2}^{2} = c^{2} + d_{1}^{2}$.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

In the following figure,if $AB = 10$,then $AC = \ldots$