MHT CET · Maths · Three Dimensional Geometry
A line makes the same angle ' \(\alpha\) ' with each of the \(\mathrm{x}\) and \(\mathrm{y}\) axes. If the angle ' \(\theta\) ', which it makes with the z-axis, is such that \(\sin ^2 \theta=2 \sin ^2 \alpha\), then the angle \(\alpha\) is
- A \(\left(\frac{\pi}{6}\right)\)
- B \(\left(\frac{\pi}{4}\right)\)
- C \(\left(\frac{\pi}{2}\right)\)
- D \(\left(\frac{\pi}{3}\right)\)
Answer & Solution
Correct Answer
(B) \(\left(\frac{\pi}{4}\right)\)
Step-by-step Solution
Detailed explanation
Direction cosines of the line are \(\cos ^2 \alpha+\cos ^2 \alpha+\cos ^2 \theta=1\)
\(\begin{aligned} & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-\sin ^2 \theta=1 \\ & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-2 \sin ^2 \alpha=1 \\ & \Rightarrow 1-\sin ^2 \alpha+1-\sin ^2 \alpha+1-2 \sin ^2 \alpha=1 \\ & \Rightarrow \sin ^2 \alpha=\frac{1}{2}=\sin ^2 \frac{\pi}{4} \\ & \Rightarrow \alpha=\frac{\pi}{4}\end{aligned}\)
\(\begin{aligned} & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-\sin ^2 \theta=1 \\ & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-2 \sin ^2 \alpha=1 \\ & \Rightarrow 1-\sin ^2 \alpha+1-\sin ^2 \alpha+1-2 \sin ^2 \alpha=1 \\ & \Rightarrow \sin ^2 \alpha=\frac{1}{2}=\sin ^2 \frac{\pi}{4} \\ & \Rightarrow \alpha=\frac{\pi}{4}\end{aligned}\)
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