Question

A flagstaff stands vertically on a pillar,the height of the flagstaff being double the height of the pillar. A man on the ground at a distance finds that both the pillar and the flagstaff subtend equal angles at his eyes. The ratio of the height of the pillar and the distance of the man from pillar, is

• A. $\sqrt{3}:1$
• B. $1:3$
• C. $1:\sqrt{3}$
• D. $\sqrt{3}:2$

Answer 1

The correct option is C $1:\sqrt{3}$

Let $AB$ be the height of flagstaff and $BC$ be the height of pillor.

$AB=2h\xi BC=h$

Now, in $△ADC$

$\Rightarrow \tan 2\theta =\frac{AC}{CD}=\frac{3h}{x}$

$\Rightarrow \tan 2\theta =\frac{3h}{x}⟶\left(1\right)$

In $\tan \theta =\frac{h}{x}⟶\left(2\right)$

From $\left(1\right),$

$\Rightarrow \tan 2\theta =\frac{3h}{x}$

$\Rightarrow \frac{2\tan \theta }{1-{\tan }^2\theta }=\frac{3h}{x}$

Using equation $\left(2\right),$

$\Rightarrow \frac{2\left(\frac{h}{x}\right)}{1-{\left(\frac{h}{x}\right)}^2}=3\frac{h}{x}$

$\Rightarrow 2=3[1-{\left(\frac{h}{x}\right)}^2]$

$\Rightarrow 2=3-{3\left(\frac{h}{x}\right)}^2$

${\Rightarrow 3\left(\frac{h}{x}\right)}^2=1$

${\Rightarrow \left(\frac{h}{x}\right)}^2=\frac{1}{3}$

$\Rightarrow \frac{h}{x}=\frac{1}{\sqrt{3}}$

$\therefore h:x=1:\sqrt{3}$

Hence, the answer is $1:\sqrt{3}.$