Two vertical poles 20 m and 80 m high
stand apart on a horizontal plane. The height of the point of
intersection of the lines joining the top of each pole to the
foot of the other is

Let$PQ$ and $RS$ be the poles of height 20 m
and 80 m subtending angles $a$and $b$at $R$and $P$respectively.
Let $h$be the height of the point $T$, the intersection of $QR$and
$PS$. (Fig. 27.12)

Then $\mid PR=h\cot \alpha +h\cot \beta =20\cot \alpha =80\cot \beta$

$\Rightarrow \cot \alpha =4\cot \beta \text{ or }\frac{\cot \alpha }{\cot \beta }=4$

Again $h \cot \alpha +h \cot \beta =20\cot \alpha$

$\begin{array}{l}\Rightarrow (h-20)\cot \alpha =-h\cot \beta \\ \Rightarrow \frac{\cot \alpha }{\cot \beta }=\frac{h}{20-h}=4\\ \Rightarrow h=80-4h\Rightarrow h=16\mathrm{m}.\end{array}$