question_answer

From an aeroplane vertically over a straight horizontally road, the angles of depression of two consecutive mile stones on opposite sides of the aeroplane are observed to be a and b, then the height in miles of aeroplane above the road is   [MNR 1986; UPSEAT 1999]

A) \[\frac{\tan \alpha \,.\,\tan \beta }{\cot \alpha +\cot \beta }\]

B) \[\frac{\tan \alpha +\tan \beta }{\tan \alpha \,.\,\tan \beta }\]

C) \[\frac{\cot \alpha +\cot \beta }{\tan \alpha \,.\,\tan \beta }\]

D) \[\frac{\tan \alpha \,.\,\tan \,\beta }{\tan \alpha +\tan \beta }\]

Correct Answer: D

Solution :

\[{{d}_{1}}=h\cot \alpha \] and \[{{d}_{2}}=h\cot \beta \] \[{{d}_{1}}+{{d}_{2}}=1\] mile = \[h(\cot \alpha +\cot \beta )\] \[\Rightarrow \] \[h=\frac{1}{(\cot \alpha +\cot \beta )}=\frac{\tan \alpha .\tan \beta }{\tan \alpha +\tan \beta }\].