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JEE MainMathematicsThree Dimensional Geometry

If the length of the perpendicular drawn from the point P( , 3, 5) , where > 2 , on the line x-1 1 = y-2 1 = z-3 1 is 2 units, and is the plane containing the point P and the given line, then the square of the perpendicular distance of the point A(5, 4, 3) from the plane is equal to

Options

  1. A6
  2. B0
  3. C14
  4. D4

Correct answer

A. 6

Step-by-step solution

Let a point on the line x-1 1 = y-2 1 = z-3 1 = be M( +1, +2, +3) . The direction ratios of the line are 1, 1, 1 . The direction ratios of the line segment joining P( , 3, 5) and M are +1- , -1, -2 . Since PM is perpendicular to the given line: 1( +1- ) + 1( -1) + 1( -2) = 0 3 - - 2 = 0 = +2 3 The square of the distance PM is 2 : ( +1- )^2 + ( -1)^2 + ( -2)^2 = 2 Substituting = +2 3 : ( 5-2 3 )^2 + ( -1 3 )^2 + ( -4 3 )^2 = 2 (25 - 20 + 4 ^2) + ( ^2 - 2 + 1) + ( ^2 - 8 + 16) = 18 6 ^2 - 30 + 42 = 18 6 ^2 - 30 + 24

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