Question

If the angle of elevation of a cloud from a point 'h' metres above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$ , prove that the height of the cloud is $h\left(\frac{tan\beta +tan\alpha }{tan\beta -tan\alpha }\right)$

Answer 1

$\begin{array}{c}\tan \alpha =\frac{H-h}{Ec}\\ Ec=\frac{H-h}{\tan \alpha }\to \left(i\right)\\ In\Delta ECF\\ \tan \beta =\frac{h+H}{Ec}\\ Ec=\frac{h+H}{\tan \beta }\to \left(ii\right)\\ \frac{H-h}{\tan \alpha }=\frac{h+H}{\tan \beta }\\ \left(h+H\right)\tan \alpha =\left(H-h\right)\tan \beta \\ h\tan \left(\tan \alpha +\tan \beta \right)=H\left(\tan \beta -\tan \alpha \right)\\ H=h\frac{\left(\tan \beta +\tan \alpha \right)}{\tan \beta -\tan \alpha }\end{array}$