Question

$P$ and $Q$ are two points on a circle of centre $C$ and radius $\alpha$, the angle $PCQ$ being $2 \theta$ then the radius of the circle inscribed the triangle $CPQ$ is maximum when -

• A. $\sin \theta=\frac{\sqrt{3}-1}{2 \sqrt{2}}$
• B. $\sin \theta=\frac{\sqrt{5}-1}{2}$
• C. $\sin \theta=\frac{\sqrt{5}+1}{2}$
• D. $\sin \theta=\frac{\sqrt{5}-1}{4}$

Answer 1

Answer: (B)

$s =\frac{2 \alpha+2 \alpha \sin \theta}{2}=\alpha+\alpha \sin \theta$
$\Delta=\frac{1}{2} \alpha^2 \sin 2 \theta$
$r =\frac{\Delta}{ s }=\frac{1}{2} \alpha\left\{\frac{\sin 2 \theta}{1+\sin \theta}\right\}$
Maximize '$r$'.