From a lighthouse the angles of depression of two ships on opposite sides of the light house are observed to be 30° and 45°. If the height of the lighthouse is h metres, the distance between the ships is

Answer: Option A

Let AB be lighthouse and P and Q are two ships on its opposite sides which form angle of elevation of A as 45° and 30° respectively AB = h
Let PB = x and QB = y

$$\begin{array}{rl}& \text{Now in right }\Delta APB\\ & \tan \theta =\frac{\text{Perpendicular}}{\text{Base}}=\frac{AB}{PB}\\ & \Rightarrow \tan 45^{\circ }=\frac{h}{x}\Rightarrow 1=\frac{h}{x}\\ & \Rightarrow x=h............(i)\\ & \text{Similarly in right }\Delta AQB,\\ & \tan 30^{\circ }=\frac{AP}{QB}=\frac{h}{y}\\ & \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{y}\\ & \Rightarrow y=\sqrt{3}h..............(ii)\\ & \text{Adding (i) and (ii)}\\ & \therefore PQ=x+y\\ & =h+\sqrt{3}h\\ & =\left(\sqrt{3}+1\right)h\end{array}$$