question_answer

A vertical tower stands on a horizontal plane and is surrounded by a vertical flagstaff of height h. At a point on the plane, the angle of a elevation of the bottom of the flagstaff is a and that of the top of the flagstaff is P. Then, the height of the tower is

A)  \[\frac{h\tan \beta }{\tan \alpha -\tan \beta }\]

B)  \[\frac{h\tan \alpha }{\tan \alpha +\tan \beta }\]

C)  \[\frac{h\tan \beta }{\tan \alpha +\tan \beta }\]

D)  \[\frac{h\tan \alpha }{\tan \beta -\tan \alpha }\]

Correct Answer: D

Solution :

\[\therefore \] Height of the light house = 225 m

Let BC be the tower, CD the flagstaff and

\[\angle BAC=\alpha \]and \[\angle BAD=\beta \]

In \[\Delta ABC,\]

\[\tan \alpha =\frac{BC}{AC}\]                           ?(i)

and in \[\Delta ABD,\]

\[\frac{BD}{AC}=\frac{BC+h}{AC}=\tan \beta \]                       ?(ii)

On dividing Eq. (ii) by Eq. (i), we get

\[\frac{BC+h}{BC}=\frac{\tan \beta }{\tan \alpha }\]

\[\Rightarrow \]   \[(BC+h)\tan \alpha =BC=\tan \beta \]

\[\Rightarrow \]   \[BC\,\,(\tan \beta -\tan \alpha )=h\tan \alpha \]

\[\Rightarrow \]   \[BC=\frac{h-\tan \alpha }{\tan \beta -\tan \alpha }\]