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Let P(2, 5, 4) be a point and L be the line x-1 1 = y-2 2 = z-3 2 . Let Q be the foot of the perpendicular drawn from P to the line L. If A and B are two points on L such that Q is the midpoint of the line segment AB and OA OB = 9 , where O is the origin, then the square of the length of the line segment AB is equal to:

Options

  1. A12
  2. B36
  3. C144
  4. D4

Correct answer

C. 144

Step-by-step solution

The equation of the line L is x-1 1 = y-2 2 = z-3 2 = . Any point on the line is given by R(1+ , 2+2 , 3+2 ) . Let Q be the foot of the perpendicular from P(2, 5, 4) to L. The direction ratios of PQ are ( -1, 2 -3, 2 -1) . Since PQ is perpendicular to L, the dot product of their direction ratios is zero: 1( -1) + 2(2 -3) + 2(2 -1) = 0 9 - 9 = 0 = 1 . Thus, the coordinates of Q are (2, 4, 5) . Since Q is the midpoint of AB, A and B are symmetric with respect to Q. Let the parameter for A be 1+t and for B be 1-t . Th

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