<p>Given lines \(\dfrac{x}{2} = \dfrac{y}{2} = \dfrac{z}{1}\) and \(\dfrac{x-5}{2} = \dfrac{y-2}{p/7} = \dfrac{z-3}{4}\). The value of \(p\) such that the angle between both lines satisfies \(\cos^{-1}\left(\dfrac{2}{3}\right)\) is:</p>
Step-by-Step Solution
Key Concept: The angle θ between two lines with direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂) is given by cos θ = |a₁a₂+b₁b₂+c₁c₂|/(√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²)). Set this equal to 2/3 and solve for p.
Step 1: Identify direction ratios. Line 1: Direction ratios are (2, 2, 1) Line 2: Direction ratios are (2, p/7, 4) Step 2: Apply the angle formula. cos θ = |a_1a_2 + b_1b_2 + c_1c_2| / (√(a_1^2 + b_1^2 + c_1^2) · √(a_2^2 + b_2^2 + c_2^2)) Step 3: Calculate the dot product. a_1a_2 + b_1b_2 + c_1c_2 = (2)(2) + (2)(p/7) + (1)(4) = 4 + 2p/7 + 4 = 8 + 2p/7 Step 4: Calculate the magnitudes. √(a_1^2 + b_1^2 + c_1^2) = √(4 + 4 + 1) = √9 = 3 √(a_2^2 + b_2^2 + c_2^2) = √(4 + p^2/49 + 16) = √(20 + p^2/49) Step 5: Set up the equation with cos⁻^1(2/3). 2/3 = |8 + 2p/7| / (3√(20 + p^2/49)) Step 6: Simplify and solve for p. 2/3 = |8 + 2p/7| / (3√(20 + p^2/49)) 2√(20 + p^2/49) = |8 + 2p/7| 4(20 + p^2/49) = (8 + 2p/7)^2 80 + 4p^2/49 = 64 + 32p/7 + 4p^2/49 80 = 64 + 32p/7 16 = 32p/7 p = 7/2 or p = 3.5 ∴ Answer: A
Correct Answer: A