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Let M be the foot of the perpendicular drawn from the point P(2, 3, -3) to the line L₁ : x-2 1 = y-1 2 = z+1 -1 . A line L₂ passes through M and is parallel to the vector 2 i + j + 2 k . If d is the shortest distance between the line L₂ and the line L₃ : x 1 = y 2 = z 2 , then the value of 17d² is equal to

Correct answer

324

Step-by-step solution

Let M be a general point on the line L₁ . Then, M = (t+2, 2t+1, -t-1) for some scalar t . The direction ratios of the line segment PM are (t+2-2, 2t+1-3, -t-1-(-3)) = (t, 2t-2, -t+2) . Since PM is perpendicular to L₁ , the dot product of the direction ratios of PM and L₁ is zero: 1(t) + 2(2t-2) - 1(-t+2) = 0 t + 4t - 4 + t - 2 = 0 6t = 6 t = 1 Thus, the coordinates of M are (3, 3, -2) . The line L₂ passes through M(3, 3, -2) and is parallel to b ₁ = 2 i + j + 2 k . The line L₃ passes through A(0, 0, 0) and is paral

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