Question

A vertical pole subtends an angle ${\tan }^{-1}\left(\frac{1}{2}\right)$ at a point P on the ground. If the angles subtended by the upper half and the lower half of the pole at P are respectively $\alpha$ and $\beta$ then $(\tan \alpha ,\tan \beta )=$

• A. $(\frac{1}{4},\frac{1}{5})$
• B. $(\frac{1}{5},\frac{2}{9})$
• C. $(\frac{2}{9},\frac{1}{4})$
• D. $(\frac{1}{4},\frac{2}{9})$

Answer 1

Answer: (C)

$(\frac{2}{9},\frac{1}{4})$

From the diagram,

$\tan (\alpha +\beta )=\frac{1}{2}=\frac{h}{d}$, $\tan \beta =\frac{h}{2d}=\frac{1}{4}$

$\therefore \frac{\tan \alpha +\frac{1}{4}}{1-\frac{\tan \alpha }{4}}=\frac{1}{2}$

$\Rightarrow \tan \alpha =\frac{2}{9}$