Question

Three cirlces are such that each touch the other two externally, The common tangets are concurrent at $P$. The length of the tangent to each circle is $p$. The ratio of the product of their radii to sum of their radii is

• A. $\frac{p}{2}$
• B. $\frac{p^2}{2}$
• C. $p$
• D. $p^2$

Answer 1

Answer: (D)

$p^2$

From the figure,
$\tan \frac{\alpha }{2}=\frac{p}{r_1},\tan \frac{\beta }{2}=\frac{p}{r_2}$
$\Rightarrow \cot \left(\frac{\alpha +\beta }{2}\right)=\frac{p}{r_3}$
$\tan \left(\frac{\alpha +\beta }{2}\right)=\frac{p(\frac{1}{r_1}+\frac{1}{r_2})}{1-\frac{p^2}{r_1{r}_2}}=\frac{p(r_1+r_2)}{\left[\right(r_1{r}_2)-p^2]}$
$\Rightarrow \frac{r_1{r}_2-p^2}{p(r_1+r_2)}=\frac{p}{r_3}$
$\Rightarrow r_1{r}_2{r}_3-r_3{p}^2=p^2(r_1+r_2)$
$\Rightarrow p^2=\frac{r_1{r}_2{r}_3}{r_1+r_2+r_3}$