Two poles of height 10 meters and 20 meters stand at the centres of...

Let, $A$ and $B$ be the centres of the two circles where the poles of height $10m$ and $20m$ respectively stand making angles ${30}^{o}$ and ${60}^{o}$ respectively at the point $O$ where these circles touch each other externally.

The sum of the radii of the two circles = The distance between the centres of the two circles.
$= 10cot {30}^{o} + 20cot {60}^{o}$
$= 10\sqrt{3} + \dfrac{20}{\sqrt{3}} = \dfrac{30 + 20}{\sqrt{3}}$& meters