Question

$A$ man observes a tower$AB$ of height $h$ from a point $P$ on the ground. He moves a distance $d$ towards the foot of the tower and finds that the angle of elevation is doubled. He further moves a distance $\frac{3d}{4}$ in the same direction and the angle of elevation is three times that at $P$. Then $\frac{h^2}{d^2}=$

• A. $\frac{35}{9}$
• B. $\frac{35}{36}$
• C. $\frac{36}{5}$
• D. $\frac{36}{35}$

Answer 1

The correct option is B $\frac{35}{36}$

In $△BQR$ apply sine formula
$\frac{d}{\sin (\pi -3\alpha )}=\frac{\frac{3}{4}d}{\sin \alpha }=4\sin \alpha -3\sin 3\alpha$
$\Rightarrow 4\sin \alpha =3(3\sin \alpha -4{\sin }^3\alpha )$
Since $\sin \alpha \ne 0,4=9-12{\sin }^2\alpha \Rightarrow {\sin }^2\alpha =\frac{5}{12}$
From $△BQA$
$h=BQ{\sin }^2\alpha =2d\sin \alpha \cos \alpha =2d\sin d\sqrt{1-{\sin }^2\alpha }$
$=2d\sqrt{\frac{5}{12}}\sqrt{1-\frac{5}{12}}=\frac{2d}{12}\sqrt{35}$
$\Rightarrow 36h^2=35d^2$

$\Rightarrow \frac{h^2}{d^2}=\frac{35}{36}$