question_answer

A tower subtends an angle \[\alpha \]at a point in the plane of its base and the angle of depression of the foot of the tower at a point b ft just above A is \[\beta .\]Then height of the tower is:

A) \[b\text{ }tan\text{ }\alpha \text{ }cot\beta \]                  

B)                  \[~b\text{ }cot\text{ }\alpha \text{ }tan\text{ }\beta \]                

C)                  \[b\text{ }tan\text{ }\alpha \text{ }tan\text{ }\beta \]                   

D)                   \[b\cot \alpha \cot \beta \]

Correct Answer: A

Solution :

Let the height of the tower be\[h.\] In \[\Delta ABC,\] \[\tan \alpha =\frac{h}{AB}\] \[\Rightarrow \]               \[AB=h\cot \alpha \]                      ?(i) In \[\Delta ABD,\] \[\tan \beta =\frac{b}{AB}\]                 \[\Rightarrow \]               \[AB=b\cot \beta \]                                        ?(ii) From Eqs.(i) and (ii) \[h=\cot \alpha =b\cot \beta \]                 \[\Rightarrow \]               \[h=b\tan \alpha \cot \beta \]