If the angles of depression of the top and bottom of a short building from the top of a tall building are $30^{\circ}$ and $60^{\circ}$ respectively, then the ratio of the heights of short and tall buildings is

• Correct answer
  A

  $2: 3$
• B

  $1: 2$
• C

  $1: 3$
• D

  $1: \mathbf{4}$

Solution

Let, $A B$ is tall building $C Q$ is short building
$\begin{aligned}<br />& A B=H \\<br />& C Q=h<br />\end{aligned}$
In $\triangle A B C$
$\begin{aligned}<br />\tan 60^{\circ} & =\frac{H}{A C} \\<br />A C & =\frac{H}{\sqrt{3}}<br />\end{aligned}$
$\begin{aligned}<br />& \text {In } \triangle B P Q \\<br />& \tan 30^{\circ}=\frac{H-h}{P Q} \\<br />& \Rightarrow \quad P Q=(H-h) \sqrt{3} \\<br />& A C=P Q \\<br />& \therefore \quad \frac{H}{\sqrt{3}}=(H-h) \sqrt{3}<br />\end{aligned}$

$\begin{aligned}<br />\Rightarrow & H =3 H-3 h \\<br />\Rightarrow & 2 H =3 h \\<br />\Rightarrow & \frac{h}{H} =\frac{2}{3} \\<br />\therefore & h: H =2: 3<br />\end{aligned}$

Hence, the correct answer is option: 'A'.