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 the pillar is

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

Answer 1

The correct option is D

$\frac{1}{\sqrt{3}}$

Explanation for the correct option:

Step1. Draw the diagram.

Here,

$\text{'}\mathrm{P}\text{'}$ is the position of the flagstaff.

$\text{'}\mathrm{S}\text{'}$ is the position of the pillar.

$\mathrm{RQ}$ is the distance between a man from the pillar.

$\mathrm{\theta }$ is the angle subtends between both flagstaff and pillar with man's eyes

Step2. Find the value of $\mathbf{tan\theta }$

In$∆\mathrm{QRS}$,

$\begin{array}{rcl}\mathrm{tan\theta }& =& \frac{\mathrm{SQ}}{\mathrm{RQ}}\mathbf{[}\mathbf{\because }\mathbf{tan\theta }\mathbf{=}\frac{\mathbf{Opposite}\mathbf{}\mathbf{side}}{\mathbf{Adjacent}\mathbf{}\mathbf{side}}\mathbf{]}\end{array}$

$\begin{array}{rcl}& \Rightarrow & \mathrm{tan\theta }=\frac{\mathrm{h}}{\mathrm{a}}\_\_\_\_\left(1\right)\end{array}$

Step3. Find the ratio of the height of the pillar and the distance of the man from the pillar

$\begin{array}{rcl}& & \mathrm{In}∆\mathrm{PQR},\end{array}$

$\begin{array}{rcl}\tan 2\theta & =& \frac{PR}{RQ}\\ \tan 2\theta & =& \frac{3h}{a}\mathbf{[}\mathbf{\because }\mathbf{PQ}\mathbf{}\mathbf{=}\mathbf{}\mathbf{PS}\mathbf{}\mathbf{+}\mathbf{SQ}\mathbf{]}\\ & \Rightarrow & \frac{2\tan \theta }{1-{\tan }^2\theta }=\frac{3h}{a}\left[\mathbf{\because }\mathbf{}\mathbf{tan}\mathbf{2}\mathbf{\theta }\mathbf{}\mathbf{=}\frac{\mathbf{2}\mathbf{tan\theta }}{\mathbf{1}\mathbf{-}{\mathbf{tan}}^{\mathbf{2}}\mathbf{\theta }}\right]\end{array}$

Substitute the value of $\mathrm{tan\theta }$ from $\left(1\right)$ we get,

$\begin{array}{rcl}\frac{2\left({\displaystyle \frac{\mathrm{h}}{\mathrm{a}}}\right)}{1-{\left({\displaystyle \frac{\mathrm{h}}{\mathrm{a}}}\right)}^2}& =& 3\left(\frac{\mathrm{h}}{\mathrm{a}}\right)\\ & \Rightarrow & \frac{2}{1-{\left({\displaystyle \frac{\mathrm{h}}{\mathrm{a}}}\right)}^2}=3\\ & \Rightarrow & 1-{\left(\frac{\mathrm{h}}{\mathrm{a}}\right)}^2=\frac{2}{3}\\ & \Rightarrow & {\left(\frac{\mathrm{h}}{\mathrm{a}}\right)}^2=\frac{1}{3}\\ & \Rightarrow & \frac{\mathrm{h}}{\mathrm{a}}=\frac{1}{\sqrt{3}}\end{array}$

Therefore, $\frac{\mathbf{H}\mathbf{e}\mathbf{i}\mathbf{g}\mathbf{h}\mathbf{t}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{}\mathbf{p}\mathbf{i}\mathbf{l}\mathbf{l}\mathbf{a}\mathbf{r}\mathbf{}}{\mathbf{D}\mathbf{i}\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{c}\mathbf{e}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{}\mathbf{m}\mathbf{a}\mathbf{n}\mathbf{}\mathbf{f}\mathbf{r}\mathbf{o}\mathbf{m}\mathbf{}\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{}\mathbf{p}\mathbf{i}\mathbf{l}\mathbf{l}\mathbf{a}}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{3}}}$

Hence, Option $\left(D\right)$ is the correct answer.