Question

From the top of a hill $h$ meters high, the angles of depressions of the top and the bottom of a pillar are $\alpha$ and $\beta$, respectively. The height (in metres) of the pillar is

• A. $h\frac{[tan\beta –tan\alpha ]}{[tan\beta ]}$
• B. $h\frac{[\tan \alpha –\tan \beta ]}{[\tan \alpha ]}$
• C. $h\frac{[\tan \alpha +\tan \beta ]}{[\tan \beta ]}$
• D. $h\frac{[\tan \alpha +\tan \beta ]}{[\tan \alpha ]}$

Answer 1

The correct option is A

$h\frac{[tan\beta –tan\alpha ]}{[tan\beta ]}$

Explanation for the correct option:

Step 1. Draw the figure according to the question:

From the figure, we have

$\tan \beta =\frac{h}{x}$ — (1)

$\tan \alpha =\frac{h_1}{x}$ — (2)

Step 2. Equate equation (1) and (2):

$\frac{h}{\tan \beta }=\frac{h1}{\tan \alpha }$

$\Rightarrow$ $h_1=h\left[\frac{\tan \alpha }{\tan \beta }\right]$

As we can see in the figure,

The height of the pillar $=h–h_1$

$=\left[h–h\left(\frac{\tan \alpha }{\tan \beta }\right)\right]$

$=h\left[1–\left(\frac{\tan \alpha }{\tan \beta }\right)\right]$

$=\mathit{h}\mathbf{}\frac{\mathbf{[}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{}\mathbf{\beta }\mathbf{}\mathbf{–}\mathbf{}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{}\mathbf{\alpha }\mathbf{]}}{\mathbf{[}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{}\mathbf{\beta }\mathbf{]}}$

Hence, Option ‘A’ is Correct.