Question

The angle of elevation of a cloud from a point h metres above the surface of a lake is $\theta$ and the angle of depression of its reflection in the lake is $\varphi$. Prove that the height of the cloud above the lake surface is : $h\left(\frac{tan\varphi +tan\theta }{tan\varphi -tan\theta }\right)$.

Answer 1

Let height of cloud from surface of lake be H.

$\therefore \frac{H-h}{AB}=\tan \theta$

And $\frac{H+h}{AB}=\tan \theta$

$\Rightarrow \frac{H-h}{H+h}=\frac{\tan \theta }{\tan \theta }$

By addendo and dividendo, we get

$\Rightarrow \frac{H-h+H+h}{H-h-H-h}=\frac{\tan \theta +\tan \theta }{\tan \theta -\tan \theta }$

$\frac{2H}{-2h}=\frac{\tan \theta +\tan \theta }{\tan \theta -\tan \theta }$

$\Rightarrow H=h\left(\frac{\tan \theta +\tan \theta }{\tan \theta -\tan \theta }\right)$