<p>If direction cosines of a line \(L\) satisfy \(l = -m - n\) and \(l^2 = m^2 + n^2\), then the angle \(\theta\) that the line makes is such that \(\cos\theta =\):</p>
Step-by-Step Solution
Key Concept: Use the constraint l² + m² + n² = 1 combined with the two given conditions to solve for individual direction cosines, then apply the geometric meaning that one direction cosine equals cos(θ) with respect to the z-axis.
Step 1: Use the fundamental constraint for direction cosines: l^2 + m^2 + n^2 = 1 Step 2: From the given condition l^2 = m^2 + n^2, substitute into the constraint: l^2 + l^2 = 1 → l^2 = 1/2 → l = ±1/√2 Step 3: From l = -m - n and l^2 = m^2 + n^2, we have: (m + n)^2 = m^2 + n^2 m^2 + 2mn + n^2 = m^2 + n^2 2mn = 0 → m = 0 or n = 0 Step 4: If m = 0: l = -n and l^2 = n^2 gives l = ±1/√2, n = ∓1/√2 This satisfies l^2 = m^2 + n^2 = 0 + 1/2 = 1/2 ✓ Step 5: If the line makes angle θ with the z-axis, then n = cos(θ). Taking n = 1/√2: cos θ = 1/√2 = √2/2 ∴ Answer: A (cos θ = 1/√2 or √2/2)
Correct Answer: A