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 $\frac{3 \pi}{4}$.