Question

The angle of elevation of the top of a tower as seen from two points $A$ & $B$ situated in the same line and at distances $p$ and $q$ respectively from the foot of the tower are complementary then the height of the tower is:

• A. $pq$
• B. $\frac{p}{q}$
• C. $\sqrt{pq}$
• D. None of these

Answer 1

The correct option is C $\sqrt{pq}$

Let the angle of elevation made at a distance of $p=$ $\alpha$
Then, angle of elevation made at a distance of $q=$ $90-\alpha$
Let the height of tower $=$ $h$
Then, $\tan \angle$ of elevation $=$ $\frac{Height}{distance}$
Thus, $\tan \alpha =\frac{h}{p}$
$\tan (90-\alpha )=\frac{h}{q}$ or $\cot \alpha =\frac{h}{q}$
Multiply both the equations,
$\tan \alpha \cot \alpha =\frac{h}{p}.\frac{h}{q}$
$\to \frac{h^2}{pq}=1$
Or, $h^2=pq$
$h=\sqrt{pq}$