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
$\text{tan}\theta =\text{tan}\alpha +\text{tan}\beta$