<p>If the straight lines \(x = 1 + s,\ y = -3 - \lambda s,\ z = 1 + \lambda s\) and \(x = \dfrac{t}{2},\ y = 1 + t,\ z = 2 - t\) with parameters \(s\) and \(t\), respectively, are co-planar then \(\lambda\) equals</p>
Step-by-Step Solution
Key Concept: Two lines are coplanar if and only if the scalar triple product of (vector between any two points on the lines) and the direction vectors equals zero. This condition is: (P₁P₂) · (d₁ × d₂) = 0.
Step 1: Identify the parametric forms Line 1: Point P_1 = (1, -3, 1), Direction d_1 = (1, -λ, λ) Line 2: Point P_2 = (0, 1, 2), Direction d_2 = (1/2, 1, -1) Step 2: Find vector P_1P_2 P_1P_2 = (0-1, 1-(-3), 2-1) = (-1, 4, 1) Step 3: Calculate d_1 × d_2 d_1 × d_2 = | i j k | = i (λ + λ) - j (-1 - λ/2) + k (1 + λ/2) |1 -λ λ| |1/2 1 -1| = (2λ, 1 + λ/2, 1 + λ/2) Step 4: Apply coplanarity condition P_1P_2 · (d_1 × d_2) = 0 (-1)(2λ) + (4)(1 + λ/2) + (1)(1 + λ/2) = 0 -2λ + 4 + 2λ + 1 + λ/2 = 0 5 + λ/2 = 0 λ = -10 ∴ Answer: A (λ = -10)
Correct Answer: A