question_answer

From a 60 meter high tower angles of depression of the top and bottom of a house are a and b respectively. If the height of the house is \[\frac{60\,\sin \,(\beta -\alpha )}{x},\] then x = 

A) \[\sin \,\,\alpha \,\,\sin \,\,\beta \]

B) \[\cos \,\,\alpha \,\,\cos \,\,\beta \]

C) \[\sin \,\,\alpha \,\,\cos \,\,\beta \]

D) \[\cos \,\,\alpha \,\,\sin \,\,\beta \]

Correct Answer: D

Solution :

\[H=d\tan \beta \] and \[H-h=d\tan \alpha \] \[\Rightarrow \] \[\frac{60}{60-h}=\frac{\tan \beta }{\tan \alpha }\] \[\Rightarrow \] \[-h=\frac{60\tan \alpha -60\tan \beta }{\tan \beta }\] \[\Rightarrow \] \[h=\frac{60\sin (\beta -\alpha )}{\cos \alpha \cos \beta \frac{\sin \beta }{\cos \beta }}\] \[\Rightarrow \] \[x=\cos \alpha \sin \beta \].