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JEE Main202126 Feb 2021Evening ShiftMathematicsThree Dimensional GeometryActual

Let L be a line obtained from the intersection of two planes x + 2 y + z = 6 and y + 2 z = 4 . If point P α , β , γ is the foot of perpendicular from 3 , 2 , 1 on L , then the value of 21 α + β + γ equals:

Options

  1. A102
  2. B142
  3. C68
  4. D136

Correct answer

A. 102

Step-by-step solution

Given, x + 2 y + z - 6 = 0       . . . 1 y + 2 z - 4 = 0       . . . 2 On solving, equations 1 and 2 , we get x - 3 z = - 2 ⇒ x = 3 z - 2 ⇒ y = 4 - 2 z Hence, Equation of the line is x + 2 3 = y - 4 - 2 = z 1 ⇒ line of intersection of two planes is x + 2 3 = y - 4 - 2 = z 1 = λ (Let) It is clear from the diagram, A P ⊥ a r to line ∴   AP ¯ . 3 i ^ - 2 j ^ + k ^ = 0 3 λ - 5 . 3 + - 2 λ + 2 - 2 + λ - 1 . 1 = 0 9 λ - 15 + 4

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