Question

A particle is thrown over a triangle from one end of a horizontal base and after grazing the vertex falls on the other end of the base. If $\alpha $ and $\beta $ be the base angles and $\theta $ be the angle of projection. Find the value of $tan$ $\theta $ .

• A. $\text{tan} \alpha + \text{tan} \beta $
• B. $\text{sin} \alpha + \text{sin} \beta $
• C. $\text{tan} \alpha + \text{sin} \beta $
• D. $\text{cos} \alpha + \text{cos} \alpha $

Answer 1

Answer: (A)

The situation is shown in diagram



$\text{tan} \alpha + \text{tan} \beta = \frac{\text{y}}{\text{x}} + \frac{\text{y}}{\text{R} - \text{x}}$
$\text{tan} \alpha + \text{tan} \beta = \frac{\text{yR}}{\text{x} \left(\text{R} - \text{x}\right)}$ ...(i)
Equation of trajectory is $\text{y} = \text{x tan } \theta \, \left[1 - \frac{\text{x}}{\text{R}}\right]$
or, $\text{tan} \theta = \frac{\text{yR}}{\text{x} \left(\text{R} - \text{x}\right)}$ ...(ii)
From Eqs. (i) and (ii), we have
$\text{tan} \theta = \text{tan} \alpha + \text{tan} \beta $