question_answer

From a point a metre above a lake the angle of elevation of a cloud \[\alpha \] and the angle of depression of its reflection is \[\beta \]. The height of the cloud is

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

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

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

D)  None of these

Correct Answer: A

Solution :

In\[\Delta CDE\],                 \[\cot \alpha =\frac{DC}{H-a}\]                                  ... (i) And in\[\Delta CDF\]                 \[\cot \beta =\frac{DC}{H+a}\]                                  ... (ii) From Eqs. (i) and (ii), we get                 \[(H+a)\cot \beta =(H-a)\cot \alpha \] \[\Rightarrow \]               \[H=\frac{a(-\cot \beta -\cot \alpha )}{\cot \beta -\cot \alpha }\] \[\Rightarrow \]               \[H=\frac{a(\cot \alpha +\cot \beta )}{\cot \alpha -\cot \beta }\]                 \[=\frac{(a\cos \alpha \sin \beta +\cos \beta \sin \alpha )}{\cos \alpha \sin \beta -\cos \beta \sin \alpha }\] \[=a\frac{\sin (\alpha +\beta )}{\sin (\beta -\alpha )}m\]