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JEE MainMathematicsThree Dimensional Geometry

Let P(-1, -8, 14) be a point and L be the line x-1 1 = y-2 2 = z-3 2 . Let A and B be two points on the line L such that the distances of A and B from P are equal to the distance of P from the origin O. Then the value of OA OB is equal to:

Options

  1. A50
  2. B225
  3. C8
  4. D-22

Correct answer

D. -22

Step-by-step solution

Let Q be the foot of the perpendicular from P(-1, -8, 14) to the line L. Any point on L is R(1+ , 2+2 , 3+2 ) . The direction ratios of PR are (2+ , 10+2 , -11+2 ) . For Q, PR is perpendicular to L: 1(2+ ) + 2(10+2 ) + 2(-11+2 ) = 0 2 + + 20 + 4 - 22 + 4 = 0 9 = 0 = 0 . Thus, Q is (1, 2, 3) . The distance squared of P from Q is: PQ^2 = (1 - (-1))^2 + (2 - (-8))^2 + (3 - 14)^2 = 4 + 100 + 121 = 225 . The distance squared of P from the origin O is: OP^2 = (-1)^2 + (-8)^2 + 14^2 = 1 + 64 + 196 = 261 . Given PA = PB =

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