Question

Two posts are $x$ metres apart and the height of one is double that of the other. If from the mid-point of the line joining their feet, an observer finds the angular elevations of their tops to be complementary, then the height (in meters) of the shorter post is

• A. $x\sqrt{2}$
• B. $\frac{x}{\sqrt{2}}$
• C. $\frac{x}{2\sqrt{2}}$
• D. $\frac{x}{4}$

Answer 1

The correct option is C $\frac{x}{2\sqrt{2}}$

Let the height of the shorter and the longer pole be $h$ and $2h$, respectively.
In $△ABM,$
$\begin{array}{cc}\tan \theta & =\frac{AB}{BM}\\ & =\frac{h}{x/2}\\ & =\frac{2h}{x}\dots \left(i\right)\end{array}$

In $△CDM,$
$\begin{array}{cc}\tan ({90}^{\circ }-\theta )& =\frac{CD}{MD}\\ \cot \theta & =\frac{2h}{x/2}\\ & =\frac{4h}{x}\dots \left(ii\right)\end{array}$
On multiplying equations $\left(i\right)$ and $\left(ii\right)$, we get
$\tan \theta \cdot \cot \theta =\frac{2h}{x}\cdot \frac{4h}{x}$
$\Rightarrow 1=\frac{8h^2}{x^2}$

$\Rightarrow h^2=\frac{x^2}{8}$

$\Rightarrow h=\frac{x}{2\sqrt{2}}$

Hence, option $C$ is the correct answer.