Question

The angle of elevation of a cloud from a height h above the level of water in a lake is $\alpha$ and the angle of depression of its image in the lake is $\beta$. Find the height of the cloud above the surface of the lake :

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

Answer 1

The correct option is C

$\frac{hsin(\alpha +\beta )}{sin(\beta -\alpha )}$

Let A be a point h metres above the lake EF and B be the position of the cloud.
Draw a line parallel to EF from A on BD at C.

But, BF = DF

Let, BC = m

so, BF = (m + h)

⇒ BF = DF = (m + h) metres

Consider $△$BAC,

AB = m cosec α ---------- (1)

and, AC = m cot $\alpha$

Consider ΔACD,
AC = (2h + m) cot β

Therefore, m cot α = (2h + m) cot $\beta$

⇒ m = $\frac{2hcot\beta }{(cot\alpha -cot\beta )}$
Therefore, (h + m) = $\frac{h+2hcot\beta }{(cot\alpha -cot\beta )}$ = $\frac{hsin(\alpha +\beta )}{sin(\beta -\alpha )}$