Question

From the top of a lighthouse AB, the angles of depression of two stations C and D on opposite sides at a distance d apart are α and β. The height of the lighthouse is

(a) \(\frac{d}{\text{cot}\,\alpha\,\text{cot}\,\beta}\)

(b) \(\frac{d}{\text{cot}\,\alpha\,-\text{cot}\,\beta}\)

(c) \(\frac{d\,\text{tan}\,\alpha\,\text{tan}\,\beta}{\text{tan}\,\alpha\,+\text{tan}\,\beta}\)

(d) \(\frac{d\,\text{cot}\,\alpha\,\text{cot}\,\beta}{\text{cot}\,\alpha\,+\text{cot}\,\beta}\)

Answer 1

(c) \(\frac{d\,\text{tan}\,\alpha\,\text{tan}\,\beta}{\text{tan}\,\alpha\,+\text{tan}\,\beta}\)

Let the height of the light house AB = h metres. 

Given ∠ACB = α, ∠ ADB = β, CD = d. 

tan α = \(\frac{AB}{BC}\)

⇒ tan α = \(\frac{h}{BC}\) ⇒ \(\frac{h}{\text{tan}\,\alpha}\)               .....(i)

In rt. Δ ABD, 

tan β = \(\frac{AB}{BD}\) ⇒ \(\frac{h}{BD}\) = tan β ⇒ BD = \(\frac{h}{\text{tan}\,\beta}\)      ......(ii)

∴ CD = BC + BD = \(\frac{h}{\text{tan}\,\alpha}\) + \(\frac{h}{\text{tan}\,\beta}\)           (Adding eqn. (i) and (ii)

⇒ d = \(\frac{h(\text{tan}\,\beta+\text{tan}\,\alpha)}{\text{tan}\,\alpha+\text{tan}\,\beta}\) = h = \(\frac{d\,\text{tan}\,\alpha\,\text{tan}\,\beta}{\text{tan}\,\alpha\,+\text{tan}\,\beta}\)