Question

The angle of elevation of a cloud from a point $h$ $m$ above the level of water in a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. Then height of the cloud above the water level.

• A. $\frac{h\sin (\beta -\alpha )}{\sin (\beta +\alpha )}$
• B. $\frac{h\sin (\beta -\alpha )}{\sin (\beta -\alpha )}$
• C. $\frac{h\sin (\alpha +\beta )}{\sin (\alpha -\beta )}$
• D. none of these

Answer 1

The correct option is A $\frac{h\sin (\beta -\alpha )}{\sin (\beta +\alpha )}$

In $\Delta ACP-$

$\tan \alpha =\frac{PC}{AC}=\frac{H-h}{x}$

$⟹x=\frac{H-h}{\tan \alpha }$

Similarly, in $\Delta AC{P}^{\prime }-$

$\tan \beta =\frac{C{P}^{\prime }}{AC}=\frac{H+h}{x}$

$⟹x=\frac{H+h}{\tan \beta }$

$⟹\frac{H-h}{\tan \alpha }=\frac{H+h}{\tan \beta }$

$⟹H=h\frac{\tan \alpha +\tan \beta }{\tan \beta -\tan \alpha }$

$⟹H=h\frac{\sin \alpha \cos \beta +\sin \beta \cos \alpha }{\sin \beta \cos \alpha -\sin \alpha \cos \beta }$

$⟹H=\frac{h\sin (\alpha +\beta )}{\sin (\beta -\alpha )}$