If a line makes angles $\tan ^{-1} \sqrt{7}, \tan ^{-1} \frac{\sqrt{5}}{\sqrt{3}}$ with $X$-axis, $Y$-axis…

1. $\frac{\pi}{2}$
2. $\frac{\pi}{6}$ or $\frac{5 \pi}{6}$
3. $\frac{\pi}{3}$ or $\frac{2 \pi}{3}$
4. $\frac{\pi}{4}$ or $\frac{3 \pi}{4}$

Solution

A line makes angle $\tan ^{-1} \sqrt{7}$ and $\tan ^{-1} \frac{\sqrt{5}}{\sqrt{3}}$ with $X$-axis and $Y$-axis respectively. So, Sc $ \begin{aligned} & \alpha=\tan ^{-1} \sqrt{7} \\ & \tan \alpha=\sqrt{7} \\ & \Rightarrow \quad \cos \alpha=\frac{1}{\sqrt{8}} \\ & \text { and } \\ & \beta=\tan ^{-1} \frac{\sqrt{5}}{\sqrt{3}} \\ & \tan \beta=\frac{\sqrt{5}}{\sqrt{3}} \\ & \Rightarrow \quad \cos \beta=\frac{\sqrt{3}}{\sqrt{8}} \\ & \end{aligned} $
Let angle make with $Z$-axis is $\gamma$. So, $ \begin{aligned} & \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 \quad \frac{4}{8}+\cos ^2 \gamma=1 \quad \Rightarrow \quad \frac{1}{2}+\cos ^2 \gamma=1 \end{aligned} $ $ \begin{array}{lr} \Rightarrow & \cos ^2 \gamma=\frac{1}{2} \\ \Rightarrow & \cos \gamma= \pm \frac{1}{\sqrt{2}} \\ \Rightarrow & \cos \gamma=\frac{1}{\sqrt{2}} \text { or } \frac{-1}{\sqrt{2}} \\ \Rightarrow & \gamma=\frac{\pi}{4} \text { or } \pi-\frac{\pi}{4} \\ \Rightarrow & \gamma=\frac{\pi}{4} \text { or } \frac{3 \pi}{4} \end{array} $ So, angle made by line with $Z$-axis is $\frac{\pi}{4}$ and $\frac{3 \pi}{4}$

Asked in: AP EAMCET 2018 (23 Apr Shift 1)

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