Let P be the foot of the perpendicular from the point A(2, , 1) on the line L₁ : x 1 = y-1 2 = z+1 -1 . Let the line L₂ : x-1 2 = y-3 1 = z+2 1 intersect the line L₁ at the point Q . If the square of the distance between P and Q is 24 , the

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Let P be the foot of the perpendicular from the point A(2, , 1) on the line L₁ : x 1 = y-1 2 = z+1 -1 . Let the line L₂ : x-1 2 = y-3 1 = z+2 1 intersect the line L₁ at the point Q . If the square of the distance between P and Q is 24 , then the product of all possible values of is :

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Correct answer: D. -20

Let the general point on L₁ be P( , 2 +1, - -1) . The direction ratios of L₁ are 1, 2, -1 . The direction ratios of AP are -2, 2 +1- , - -2 . Since AP L₁ , we have : 1( -2) + 2(2 +1- ) - 1(- -2) = 0 - 2 + 4 + 2 - 2 + + 2 = 0 6 + 2 - 2 = 0 = 3 + 1 Let the general point on L₂ be (2 +1, +3, -2) . For the intersection point Q of L₁ and L₂ : = 2 +1 2 +1 = +3 Solving these, we get = 0 and = 1 . Thus, Q is (1, 3, -2) . Given (PQ)^2 = 24 , we have : ( -1)^2 + (2 +1-3)^2 + (- -1+2)^2 = 24 ( -1)^2 + 4( -1)^2 + ( -1)^2 = 24 6

Let P be the foot of the perpendicular from the point A(2, , 1) on the line L₁ : x 1 = y-1 2 = z+1 -1 . Let the line L₂ : x-1 2 = y-3 1 = z+2 1 intersect the line L₁ at the point Q . If the square of the distance between P and Q is 24 , then the product of all possible values of is :

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