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Let a line L having direction ratios 2, 1, 2 intersect the lines L₁: x-1 2 = y+1 1 = z-2 3 and L₂: x- 1 = y-2 2 = z-1 1 at the points A and B respectively. If the length of the line segment AB is 15 , then the sum of the squares of all possible values of is equal to

Correct answer

20

Step-by-step solution

Let the coordinates of A on L₁ be (2 +1, -1, 3 +2) and B on L₂ be ( + , 2 +2, +1) . The vector AB is given by: AB = + -2 -1, 2 - +3, -3 -1 The direction ratios of the line L are 2, 1, 2 . The magnitude of the vector with these components is 2^2 + 1^2 + 2^2 = 3 . Since the length of AB is 15 , the vector AB must be 15 3 2, 1, 2 = 10, 5, 10 . Case 1: AB = 10, 5, 10 + -2 -1 = 10 2 - +3 = 5 2 - = 2 -3 -1 = 10 -3 = 11 Solving the last two equations for and : Multiply the second equation by 2 : 2 - 6 = 22 . Subtracting t

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