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The line of shortest distance between the lines x - 1 2 = y - 1 -2 = z - 1 and x - 2 2 = y - 3 = z - 3 -2 meets them at points P and Q respectively. If a point R( , , ) lies on the line segment PQ such that PR = 2RQ , then the value of 9( + + ) is

Correct answer

57

Step-by-step solution

Let the given lines be L₁: x - 1 2 = y - 1 -2 = z - 1 1 and L₂: x - 2 2 = y - 3 1 = z - 3 -2 . The line of shortest distance is perpendicular to both L₁ and L₂ . Its direction vector n is given by the cross product of their direction vectors: n = (2 i - 2 j + k ) (2 i + j - 2 k ) = 3 i + 6 j + 6 k The direction ratios of the line of shortest distance are proportional to 1, 2, 2 . Let general points on L₁ and L₂ be P(2t+1, -2t+1, t+1) and Q(2s+2, s+3, -2s+3) . The direction ratios of PQ are (2s - 2t + 1, s + 2t + 2,

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