Question

Let PQ and RS be tangents at extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then $br=\sqrt{PQ\times RS}$. Find b

Answer 1

From Fig., we have
$\frac{PQ}{PR}=\tan (\frac{\pi }{2}-\theta )=\cot \theta$ and $\frac{RS}{PR}=\tan \theta$
or $\frac{PQ}{PR}\times \frac{RS}{PR}=1$
or ${\left(PR\right)}^2=PQ\times PS$
or ${\left(2r\right)}^2=PQ\times PS$
or $2r=\sqrt{PQ\times PS}$
Ans: 2