<p>A line makes an angle $\theta$ both with X and Y-axes. A possible value of $\theta$ is in</p>
<p>(a) $\left[0, \frac{\pi}{4}\right]$</p>
<p>(b) $\left[0, \frac{\pi}{2}\right]$</p>
<p>(c) $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$</p>
<p>(d) $\left[\frac{\pi}{3}, \frac{\pi}{6}\right]$</p>
Step-by-Step Solution
Key Concept: If a line makes angles α, β, γ with the X, Y, Z axes respectively, then cos²α + cos²β + cos²γ = 1. When a line makes equal angles θ with both X and Y axes in 3D space, we must use this constraint to find the valid range of θ.
Step 1: Let the line make angles α = θ, β = θ, and γ with the X, Y, and Z axes respectively. Step 2: By the fundamental property of direction cosines:
$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$
$\cos^2\theta + \cos^2\theta + \cos^2\gamma = 1$
$2\cos^2\theta + \cos^2\gamma = 1$ Step 3: Since $\cos^2\gamma \geq 0$, we have:
$2\cos^2\theta \leq 1$
$\cos^2\theta \leq \frac{1}{2}$
$|\cos\theta| \leq \frac{1}{\sqrt{2}}$ Step 4: For $\theta \in [0, \pi]$ (the valid range for angles with coordinate axes):
$\cos\theta \leq \frac{1}{\sqrt{2}}$
This gives: $\theta \geq \frac{\pi}{4}$ Step 5: Also, since θ is an angle with an axis: $\theta \in [0, \frac{\pi}{2}]$ Step 6: Combining both constraints:
$\theta \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$ ∴ Answer: c
Correct Answer: c