Question

$PQ$ is a post of height $\alpha ,$ and $A$ is a tower at some distance; $\alpha$ and $\beta$ are the angles of elevation of $B$, the top of tower, at $P$ and $Q$ respectively. The height of the tower is

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

Answer 1

Answer: (A)

$\frac{a\sin \alpha \cos \beta }{\sin (\alpha -\beta )}$

Let $AB=h$ be the height of tower at a distance $x$ from the post of $PQ=a$.

$AB$ subtends angle $\alpha$ and $\beta$ at $P$ and $Q$ respectively.

We have to determine $h$ and $x$ in terms of known quantities $\alpha ,\beta$ and $a$.

$AB=x\tan \alpha ;BR=x\tan \beta \therefore PQ=AR=AB-BR=x(\tan \alpha -\tan \beta )$

$\therefore x=\frac{PQ}{\tan \alpha -\tan \beta }=\frac{a}{\tan \alpha -\tan \beta }=\frac{a\cos \alpha \cos \beta }{\sin (\alpha -\beta )}$

$h=AB=x\tan \alpha =\frac{a\cos \alpha \cos \beta }{\sin (\alpha -\beta )}.\frac{\sin \alpha }{\cos \alpha }=\frac{a\sin \alpha \cos \beta }{\sin (\alpha -\beta )}$