Question

Two equal circles of radius R are touching each other externally. If a smaller circle of radius r is touching both of these circles as well as their direct common tangent then the ratio r : R is

Answer 1

Length of common tangent to circles of equal radius = $2\times$ radius

Here $l=2R$

$\Rightarrow AD=DC=BE=\frac{1}{2}=R$

$AB=R+r$

$AE=R-r$

$A{B}^2=A{E}^2+B{E}^2$

$\Rightarrow (R+r{)}^2=(R-r{)}^2+R^2$

$\Rightarrow 4Rr=R^2$

$\Rightarrow \frac{4}{1}=\frac{R}{r}$

$\Rightarrow r:R=1:4$

Hence solved.