<p>If the angle \(\theta\) between the line \(\dfrac{x+1}{1} = \dfrac{y-1}{2} = \dfrac{z-2}{2}\) and the plane \(2x - y + \sqrt{\lambda}z + 4 = 0\) is such that \(\sin\theta = \dfrac{1}{3}\), the value of \(\lambda\) is</p>
Step-by-Step Solution
Key Concept: The angle θ between a line and plane satisfies sin θ = |a·n|/(|a||n|), where a is the direction vector of the line and n is the normal vector of the plane. Use this formula directly, not the complement angle formula.
Step 1: Identify the direction vector of the line and normal vector of the plane. Direction vector of line: a = (1, 2, 2) Normal vector of plane: n = (2, -1, √λ) Step 2: Apply the sine formula for angle between line and plane. sin θ = | a · n | / (| a | | n |) Step 3: Calculate the dot product and magnitudes. a · n = (1)(2) + (2)(-1) + (2)(√λ) = 2 - 2 + 2√λ = 2√λ | a | = √(1^2 + 2^2 + 2^2) = √9 = 3 | n | = √(4 + 1 + λ) = √(5 + λ) Step 4: Substitute into the sine formula. 1/3 = |2√λ| / (3√(5 + λ)) Step 5: Solve for λ. 1/3 = 2√λ / (3√(5 + λ)) √(5 + λ) = 2√λ 5 + λ = 4λ 5 = 3λ ∴ λ = 5/3
Correct Answer: A