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 metres) 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

Answer: (C)

Let the height of the shorter and the longer pole be $h$ and $2h$ , respectively.
In $\Delta ABM,$
$\tan \theta =\frac{AB}{BM}=\frac{h}{x/2}=\frac{2h}{x}$ ... (i)
In $\Delta CDM,$ $\tan ({90}^o-\theta )=\frac{CD}{MD}$
$=\frac{2h}{x/2}=\frac{4h}{x}$
$\Rightarrow$ $\cot \theta =\frac{4h}{x}$ ... (ii)
On multiplying both the equations, we get
$1=\frac{2h}{x}\cdot \frac{4h}{x}$
$\Rightarrow$ $\frac{x^2}{8}=k^2$
$\Rightarrow$ $h=\frac{x}{2\sqrt{2}}$