Question

A line makes acute angles $\alpha ,\beta ,\gamma$ with the coordinate axes such that $\cos \alpha \cos \beta =\cos \beta \cos \gamma =\frac{2}{9}$ and $\cos \gamma \cos \alpha =\frac{4}{9}$ then $\cos \alpha +\cos \beta +\cos \gamma$ value is

• A. $\frac{25}{9}$
• B. $\frac{5}{9}$
• C. $\frac{5}{3}$
• D. $\frac{2}{3}$

Answer 1

The correct option is C $\frac{5}{3}$

Que :$cos\alpha .cos\beta =cos\beta .cos\gamma =\frac{2}{9},cos\gamma .cos\alpha =\frac{4}{5}$

$cos\alpha +cos\beta +cos\gamma =?$

$\to cos\alpha .cos\beta =\frac{2}{9}$__ (i)

$cos\beta .cos\gamma =\frac{2}{9}$__ (ii)

taking ratio of equation (i) & (ii)

$\frac{cos\alpha .cos\beta }{cos\beta .cos\gamma }=\frac{2/9}{2/9}$

$\therefore \frac{cos\alpha }{cos\gamma }=1$

$\therefore cos\alpha =cos\gamma$

$\to$ substitute value of $cos\alpha$ is $cos\gamma .cos\alpha =\frac{4}{9}$

$\therefore cos\alpha .cos\alpha =\frac{4}{9}$

$\therefore co{s}^2\alpha =\frac{4}{9}$

$\therefore cos\alpha =2/3$

$\therefore cos\gamma =2/3$

$\to$ substitute value of $cos\alpha$ & $cos\gamma$ in equation (i)

$\therefore cos\alpha .cos\beta =2/9$

$\therefore 2/3.cos\beta =2/9$

$\therefore cos\beta =\frac{2}{9}\times 3/2$

$cos\beta =\frac{1}{3}$

$\to$ So, $cos\alpha +cos\beta +cos\gamma =\frac{2}{3}+\frac{1}{3}+\frac{2}{3}$

$=\frac{2+1+2}{3}$

$=\frac{5}{3}$