Question

A tower $T_1$ of height $60$ m is located exactly opposite to a tower $T_2$ of height $80m$ on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_2$, then the width (in $m$) of the road between the feet of the towers $T_1$ and $T_2$ is:

• A. $10\sqrt{3}$
• B. $10\sqrt{2}$
  
• C. $20\sqrt{2}$
• D. $20\sqrt{3}$

Answer 1

The correct option is C $20\sqrt{2}$

Let the $\angle BAC=\theta$ then $\angle DBE=2\theta$ as shown in below figure
From $△BDO$
$\tan 2\theta =\frac{80}{w}=\frac{2\tan \theta }{1-{\tan }^2\theta }...\left(1\right)$
From $△BAC$
$\tan \theta =\frac{20}{w}$
From equation $\left(1\right)$
$\frac{2\times \frac{20}{w}}{1-{\left(\frac{20}{w}\right)}^2}=\frac{80}{w}$
$\Rightarrow w^2=800$
$\Rightarrow w=20\sqrt{2}$