Question

A straight line is equally inclined to all the three coordinate axes. Then the acute angle made by the line with the $y$-axis is:

• A. ${\cos }^{-1}\left(\frac{1}{3}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
• C. ${\cos }^{-1}\left(\frac{2}{\sqrt{3}}\right)$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is B ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

For line to be equally inclined to the coordinate axis,
$l=m=n$
i.e. $\cos \alpha =\cos \beta =\cos \gamma$
So, $\Rightarrow {\cos }^2\beta +{\cos }^2\beta +{\cos }^2\beta =1$
$\Rightarrow 3{\cos }^2\beta =1$
$\Rightarrow \cos \beta =\frac{1}{\sqrt{3}}$
$\Rightarrow \beta ={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$