Question

From the top of a lighthouse, the angles of depression of two stations on the oposite sides of it at a distance d apart are $\alpha$ and $\beta$. The height of the lighthouse is

• A. $\frac{d \, \tan \, \alpha}{\tan \, \alpha + \tan \, \beta}$
• B. $\frac{d}{\cot \, \alpha + \cot \, \beta}$
• C. $\frac{d \, \tan \, \beta}{\tan \, \alpha + \tan \, \beta}$
• D. $\frac{d \, \cot \, \beta}{\cot \, \alpha + \cot \, \beta}$

Answer 1

Answer: (B)

$\frac{d}{\cot \, \alpha + \cot \, \beta}$

Answer 2

Let $PM$ be lighthouse.

Given,
$\angle P Q M=\alpha, \angle P R M=\beta$ and $Q R=d$
In $\Delta PQM$,
$tan\,\alpha=\frac{PM}{QM}$
$\Rightarrow QM=PM\,cot\,\alpha$
In $\Delta PRM$,
$\tan\,\beta=\frac{PM}{RM}$
$\Rightarrow RM = PM\,\cot\,??$\therefore QR=QM+MR$$\Rightarrow d=PM,cot,\alpha+PM, \cot,\beta$$\Rightarrow d=PM\left(\cot,\alpha + \cot,\beta\right)$$\Rightarrow d=PM\left(\cot,\alpha + \cot,\beta\right)$$\Rightarrow PM=\frac{d}{\cot,\alpha + \cot,\beta}$$\therefore$Height of light house is$\frac{d}{\cot,\alpha + \cot,\beta}$.