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JEE Main202227 Jun 2022Evening ShiftMathematicsThree Dimensional GeometryActual

Let the foot of the perpendicular from the point 1 , 2 , 4 on the line x + 2 4 = y - 1 2 = z + 1 3 be P . Then the distance of P from the plane 3 x + 4 y + 12 z + 23 = 0 is

Options

  1. A50 13
  2. B63 13
  3. C65 13
  4. D4

Correct answer

C. 65 13

Step-by-step solution

Let P be foot of perpendicular of point Q 1 , 2 , 4 Let x + 2 4 = y - 1 2 = z + 1 3 = λ So x = 4 λ - 2 ,   y = 2 λ + 1 , z = 3 λ - 1 Then coordinates of point P will be 4 λ - 2 ,   2 λ + 1 , 3 λ - 1 Now direction ratios of Q P = 4 λ - 2 - 1 ,   2 λ + 1 - 2 ,   3 λ - 1 - 4 = 4 λ - 3 , 2 λ - 1 , 3 λ - 5 and D.R's of line will be 4 , 2 , 3 Now P Q and line are perpendicular so, 4 4 λ - 3 + 2 2 λ - 1 + 3 3 λ - 5 = 0

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