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

• A. $1:\sqrt{2}$
• B. $1:2$
• C. $1:2\sqrt{2}$
• D. $1:4$

Answer 1

The correct option is D $1:4$

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

Here $l=2R$

$\implies AD = DC = BE = \dfrac{1}{2} = R$

$AB=R+r$

$AE=R-r$

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

$⟹(R+r{)}^2=(R-r{)}^2+R^2$

$⟹4Rr=R^2$

$⟹\frac{4}{1}=\frac{R}{r}⟹r:R=1:4$