Question

If a line makes angles ${\tan }^{-1}\sqrt{7},{\tan }^{-1}\frac{\sqrt{5}}{\sqrt{3}}$ with $X$ -axis, $Y$ -axis respectively, then the angle made by it with $Z$ -axis is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{6}$ or $\frac{5\pi }{6}$
• C. $\frac{\pi }{3}$ or $\frac{2\pi }{3}$
• D. $\frac{\pi }{4}$ or $\frac{3\pi }{4}$

Answer 1

Answer: (D)

A line makes angle ${\tan }^{-1}\sqrt{7}$ and ${\tan }^{-1}\frac{\sqrt{5}}{\sqrt{3}}$ with
$X$ -axis and $Y$ -axis respectively.
So, $\alpha ={\tan }^{-1}\sqrt{7}$
$\tan \alpha =\sqrt{7}$
$\Rightarrow \cos \alpha =\frac{1}{\sqrt{8}}$
and $\beta ={\tan }^{-1}\frac{\sqrt{5}}{\sqrt{3}}$
$\tan \beta =\frac{\sqrt{5}}{\sqrt{3}}$
$\Rightarrow \cos \beta =\frac{\sqrt{3}}{\sqrt{8}}$
Let angle make with $Z$ -axis is $\gamma$. So,
${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\Rightarrow {\left(\frac{1}{\sqrt{8}}\right)}^2+\left(\frac{\sqrt{3}}{\sqrt{8}}\right)+{\cos }^2\gamma =1$
$\Rightarrow \frac{4}{8}+{\cos }^2\gamma =1$
$\Rightarrow \frac{1}{2}+{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\gamma =\frac{1}{2}$
$\Rightarrow \cos \gamma =\pm \frac{1}{\sqrt{2}}$
$\Rightarrow \cos \gamma =\frac{1}{\sqrt{2}}$ or $\frac{-1}{\sqrt{2}}$
$\Rightarrow \gamma =\frac{\pi }{4}$ or $\pi -\frac{\pi }{4}$
$\Rightarrow \gamma =\frac{\pi }{4}$ or $\frac{3\pi }{4}$
So, angle made by line with $Z$ -axis is $\frac{\pi }{4}$ and $\frac{3\pi }{4}$.