The top of two towers of heights x and y standing on level ground, subtend angles of 60° and 30° respectively at the midpoint of the line joining their feet. The value of x : y is:

Correct option is D

Given:

Two towers of heights x and y, subtending angles of 60° and 30° respectively at the midpoint of the line joining their feet.

Concept Used:

Tan function in a right triangle: $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$​​

Let the distance from the midpoint to each tower's base be d. $$\tan 60^\circ = \frac{x}{d}, \quad \tan 30^\circ = \frac{y}{d}$$​​

Values of standard tangents:  $$\tan 60^\circ = \sqrt{3}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}}$$  

Solution:

​$$\begin{aligned}&\text{From the given angles,} \\&\tan 60^\circ = \frac{x}{d} \Rightarrow x = \sqrt{3} d \\&\tan 30^\circ = \frac{y}{d} \Rightarrow y = \frac{d}{\sqrt{3}}\end{aligned}$$​​

Taking the ratio x : y,

​$$\frac{x}{y} = \frac{\sqrt{3} d}{\frac{d}{\sqrt{3}}} = \sqrt{3} \times \sqrt{3} = 3$$​​

Thus,

x:y=3:1

Option (d) is right.