Question

From the top of a hill $h$ metres high the angles of depressions of the top and the bottom of a pillar are $\alpha$ and $\beta$ respectively. The height (in metres) of the pillar is

• A. $\frac{h(\tan \beta -\tan \alpha )}{\tan \beta }$
• B. $\frac{h(\tan \alpha -\tan \beta )}{\tan \alpha }$
• C. $\frac{h(\tan \beta +\tan \alpha )}{\tan \beta }$
• D. $\frac{h(\tan \beta +\tan \alpha )}{\tan \alpha }$

Answer 1

Answer: (A)

Let $AB$ be a hill whose height is $h$ metres and $CD$ be a pillar of height $h^{{}^{\prime }}$ metres. In $\Delta EDB$
$\tan \alpha =\frac{h-h^{{}^{\prime }}}{ED}...(i)$
and in $\Delta ACB$,

$\tan \beta =\frac{h}{AC}=\frac{h}{ED}...(ii)$
Eliminate ED from Eqs. (i) and (ii),
we get
$\tan \alpha =\frac{h-h^{{}^{\prime }}}{\frac{h}{\tan \beta }}$
$\Rightarrow h\frac{\tan \alpha }{\tan \beta }=h-h^{{}^{\prime }}$
$\Rightarrow h^{{}^{\prime }}=\frac{h(\tan \beta -\tan \alpha )}{\tan \beta }$