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Let Q be the image of the point A(2, 4, -1) in the line x + 5 1 = y + 3 4 = z - 6 -9 . The perpendicular distance of the point Q from the plane x + 2y - 2z = 5 is

Options

  1. A3
  2. B1 3
  3. C9
  4. D7 3

Correct answer

A. 3

Step-by-step solution

Let P be the foot of the perpendicular from A(2, 4, -1) to the given line. The equation of the line is x + 5 1 = y + 3 4 = z - 6 -9 = . Any point on the line is of the form ( - 5, 4 - 3, -9 + 6) . Let this be P . The direction ratios of AP are ( - 5 - 2, 4 - 3 - 4, -9 + 6 - (-1)) = ( - 7, 4 - 7, -9 + 7) . Since AP is perpendicular to the line, the dot product of their direction ratios is zero: 1( - 7) + 4(4 - 7) - 9(-9 + 7) = 0 - 7 + 16 - 28 + 81 - 63 = 0 98 - 98 = 0 = 1 Substituting = 1 , the coordinates of P are

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