Q61 / 180 · KCET 2023 · Mathematics

If a line makes an angle of $\frac{\pi}{3}$ with each $X$ and $Y$ axis, then the acute angle made by $\mathrm{Z}$-axis is

  • A
    $\frac{\pi}{3}$
  • B
    $\frac{\pi}{2}$
  • Correct answer
    C
    $\frac{\pi}{4}$
  • D
    $\frac{\pi}{6}$

Solution

Given, $\alpha=\beta=\frac{\pi}{3}$ Let acute angle made by $Z$-axis be $\gamma$. We know that $\begin{aligned} & \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\ & \Rightarrow \quad\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^2+\cos ^2 \gamma=1 \\ & \Rightarrow \quad \frac{1}{4}+\frac{1}{4}+\cos ^2 \gamma=1 \\ & \Rightarrow \quad \cos ^2 \gamma=1-\frac{1}{2} \Rightarrow \cos \gamma= \pm \frac{1}{\sqrt{2}} \\ & \Rightarrow \quad \gamma=\frac{\pi}{4}\end{aligned}$ [ $\because y$ is acute]

Hence, the correct answer is option: 'C'.

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