question_answer

The height of a tower is h and the angle of elevation of the top of the tower is \[\alpha .\] On moving a distance h/2 towards the tower, the angle of elevation becomes \[\beta .\] What is the value of \[(\cot \alpha -\cot \beta )\]?

A)  \[\frac{1}{2}\]

B)  \[\frac{2}{3}\]     

C)  1

D)  2

Correct Answer: A

Solution :

[a] \[\tan \beta =\frac{h}{CD}\]\[\Rightarrow \]\[CD=\frac{h}{\tan \beta }=h\cot \beta \] and \[\tan \alpha =\frac{h}{\frac{h}{2}+CD}\]

\[\Rightarrow \]   \[\frac{h}{2}+CD=\frac{h}{\tan \alpha }=h\cot \alpha \]

\[\Rightarrow \]   \[\frac{h}{2}+h\,\,\cot \beta =h\,\,\cot \alpha \]

\[\Rightarrow \]\[h\cot \alpha -h\cos \beta =\frac{h}{2}\]\[\Rightarrow \]\[(\cot \alpha -\cot \beta )=\frac{h}{2}\]

\[\therefore \]      \[\cot \alpha -cot\beta =\frac{1}{2}\]