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Let a line L passing through the point P(2, 3, 1) intersect the lines L₁: x = y-1 = z-2 and L₂: x-1 2 = y-5 -1 = z+4 3 at the points A and B respectively. Then AB^2 is equal to

Correct answer

24

Step-by-step solution

Let the coordinates of A on L₁ be ( , +1, +2) and B on L₂ be (2 +1, - +5, 3 -4) . Since the line L passes through P(2, 3, 1) , A , and B , the points P, A, B are collinear. Thus, the vectors PA and PB are proportional. PA = -2, -2, +1 PB = 2 -1, - +2, 3 -5 Using the condition of proportionality: -2 2 -1 = -2 - +2 = +1 3 -5 From the first two ratios, either - 2 = 0 or 2 - 1 = - + 2 . If = 2 , then PA = 0, 0, 3 , which implies 2 - 1 = 0 and - + 2 = 0 , giving inconsistent values for . Thus, 2 - 1 = - + 2 3 = 3 = 1 .

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