Two poles of height $a$and $b$stand at the
centres of two circular plots which touch each other externally
at a point and the two poles subtend angles of 30° and
60° respectively at this point, then distance between the
centres of these poles is

Let $A$and $B$be the centres of the two circles
where the poles of height $a$and $b$respectively stand making
angles 30° and 60° respectively at the point $O$where these
circle touch each other externally. (Fig. 27.33)

sum of the radii of the two circles

$\begin{array}{l}=a\cot {30}^{\circ }+b\cot {60}^{\circ }\\ =a\sqrt{3}+\frac{b}{\sqrt{3}}=\frac{3a+b}{\sqrt{3}}\end{array}$

= distance between the centres of the two circles.