Question

If $A,A_1,A_2,A_3$ be the area of the in-circle and ex-circles, then $\frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}$ is equal to

• A. $\frac{1}{\sqrt{A}}$
• B. $\frac{2}{\sqrt{A}}$
• C. $\frac{3}{\sqrt{A}}$
• D. $None$

Answer 1

The correct option is A $\frac{1}{\sqrt{A}}$

$A_1=\pi {r_1}^2=\frac{\pi {\Delta }^2}{(s-a{)}^2}$

$A_2=\pi {r_2}^2=\frac{\pi {\Delta }^2}{(s-b{)}^2}$

$A_3=\pi {r_3}^2=\frac{\pi {\Delta }^2}{(s-c{)}^2}$

$A=\pi r^2=\frac{\pi {\Delta }^2}{(s{)}^2}$

$\frac{1}{\sqrt{A_1}}+\frac{1}{\sqrt{A_2}}+\frac{1}{\sqrt{A_3}}=\frac{1}{\sqrt{\pi }}[\frac{s-a}{\Delta }+\frac{s-b}{\Delta }+\frac{s-c}{\Delta }]=\frac{1}{\sqrt{\pi }\Delta }[3s-(a+b+c\left)\right]=\frac{s}{\sqrt{\pi }\Delta }=\frac{1}{\sqrt{A}}$