Question

If $A,A_1,A_2,A_3$ be the areas of the incircle and exircles, then $\frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}=\frac{k}{\sqrt{A}}$, then $k=$

Answer 1

Area of a circle $=\pi \times {\left(radius\right)}^2$

$\therefore A=\pi r^2,A_1=\pi {r_1}^2,A_2=\pi {r_2}^2,A_3=\pi {r_3}^2$

$\therefore \frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}=\frac{1}{r_1\sqrt{\pi }}+\frac{1}{r_2\sqrt{\pi }}+\frac{1}{r_3\sqrt{\pi }}$

$=\frac{1}{\sqrt{\pi }}[\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}]=\frac{1}{\sqrt{\pi }}[\frac{s-a}{\Delta }+\frac{s-b}{\Delta }+\frac{s-c}{\Delta }]$

$=\frac{1}{\sqrt{\pi }}\left[\frac{3s-(a+b+c)}{\Delta }\right]=\frac{1}{\sqrt{\pi }}.\frac{3s-2s}{\Delta }=\frac{1}{\sqrt{\pi }}.\frac{s}{\Delta }$

$=\frac{1}{r\sqrt{\pi }}=\frac{1}{\sqrt{\pi r^2}}=\frac{1}{\sqrt{A}}$