question_answer

From a point a metre above a lake the angle of elevation of a cloud is a and the angle of depression of its reflection is b. The height of the cloud is [Roorkee 1983; EAMCET 1983, 85]

A) \[\frac{a\sin \,(\alpha +\beta )}{\sin \,(\alpha -\beta )}\] metre

B) \[\frac{a\sin \,(\alpha +\beta )}{\sin \,(\beta -\alpha )}\] metre

C) \[\frac{a\sin \,(\beta -\alpha )}{\sin \,(\alpha +\beta )}\] metre

D) None of these

Correct Answer: B

Solution :

\[(H+a)\cot \beta =(H-a)\cot \alpha \] \[H=\frac{a\sin (\alpha +\beta )}{\sin (\beta -\alpha )}\] Using \[\cot \alpha +\cot \beta =\frac{\sin (\alpha +\beta )}{\sin \alpha \sin \beta }\] and \[\cot \alpha -\cot \beta =\frac{\sin (\beta -\alpha )}{\sin \alpha \sin \beta }\].