question_answer

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) \[\frac{x}{4}\]

Correct Answer: C

Solution :

Let the height of the shorter and the longer pole be \[h\] and\[2h\], respectively. e 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                 \[1=\frac{2h}{x}\cdot \frac{4h}{x}\] \[\Rightarrow \]               \[\frac{{{x}^{2}}}{8}={{k}^{2}}\] \[\Rightarrow \]               \[h=\frac{x}{2\sqrt{2}}\]