Fill in the blanks:
$(i)$ $A$ circle can have $......$ parallel tangents at the most.
$(ii)$ The common point of a tangent to a circle and the circle is called $......$

$A$ triangle $ABC$ is drawn to circumscribe a circle of radius $4 \, cm$ such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $8 \, cm$ and $6 \, cm$ respectively (see figure). Find the sides $AB$ and $AC$ (in $cm$).

From a point $Q$,the length of the tangent to a circle is $24 \, cm$ and the distance of $Q$ from the centre is $25 \, cm$. The radius of the circle is (in $cm$)

$A$ tangent $PQ$ at a point $P$ of a circle of radius $5 \, cm$ meets a line through the centre $O$ at a point $Q$ so that $OQ = 12 \, cm$. The length of $PQ$ is (in $cm$):

In $Fig.$,$XY$ and $X'Y'$ are two parallel tangents to a circle with centre $O$ and another tangent $AB$ with point of contact $C$ intersecting $XY$ at $A$ and $X'Y'$ at $B$. Prove that $\angle AOB = 90^{\circ}$.

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.