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

Answer 1

The correct option is A $\frac{h\tan \beta -h\tan \alpha }{\tan \beta }$

Let $AB$ be a hill whose height is $h$ metres and $CD$ be a pillar of height $h$ 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)
Divide equations (i) and (ii) to eliminate $ED$:
$\frac{\tan \alpha }{\tan \beta }=\frac{h-h^{\prime }}{h}$

$h.\frac{\tan \alpha }{\tan \beta }=h-h^{\prime }$

$⟹h^{\prime }=\frac{h\tan \beta -h\tan \alpha }{\tan \beta }$