Question

Choose the correct answer of the following question:

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

(a) 1 : 2 (b) 2 : 1 (c) 1 : 3 (d) 3 : 1 [CBSE 2015]

Answer 1

Let AB and CD be the two towers such that AB = x and CD = y.

We have,

$$\begin{gathered} \angle \mathrm{AEB}=30°,\angle \mathrm{CED}=60°\mathrm{and}\mathrm{BE}=\mathrm{DE} \\ \mathrm{In}∆\mathrm{ABE}, \\ \tan 30°=\frac{\mathrm{AB}}{\mathrm{BE}} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{x}{\mathrm{BE}} \\ \Rightarrow \mathrm{BE}=x\sqrt{3} \\ \mathrm{Also},\mathrm{in}∆\mathrm{CDE}, \\ \tan 60°=\frac{\mathrm{CD}}{\mathrm{DE}} \\ \Rightarrow \sqrt{3}=\frac{y}{\mathrm{DE}} \\ \Rightarrow \mathrm{DE}=\frac{y}{\sqrt{3}} \\ \mathrm{As},\mathrm{BE}=\mathrm{DE} \\ \Rightarrow x\sqrt{3}=\frac{y}{\sqrt{3}} \\ \Rightarrow \frac{x}{y}=\frac{1}{\sqrt{3}\times \sqrt{3}} \\ \Rightarrow \frac{x}{y}=\frac{1}{3} \\ \therefore x:y=1:3 \end{gathered}$$

Hence, the correct answer is option (c).