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JEE Main202120 Jul 2021Evening ShiftMathematicsThree Dimensional GeometryActual

Consider the line L given by the equation x - 3 2 = y - 1 1 = z - 2 1 . Let Q be the mirror image of the point ( 2 , 3 , - 1 ) with respect to L . Let a plane P be such that it passes through Q , and the line L is perpendicular to P . Then which of the following points is on the plane P ?

Options

  1. A- 1 ,   1 ,   2
  2. B1 ,   1 ,   1
  3. C1 ,   1 ,   2
  4. D1 ,   2 ,   2

Correct answer

D. 1 ,   2 ,   2

Step-by-step solution

We have, L : x - 3 2 = y - 1 1 = z - 2 1 = k Let R ≡ 2 , 3 , - 1 and Q ≡ a , b , c And, M ≡ 2 k + 3 , k + 1 , k + 2 . Direction ratios of R M is 2 k + 1 , k - 2 , k + 3 , Then, R M ⊥ L , therefore 2 2 k + 1 + k - 2 + k + 3 = 0 ⇒ 6 k = - 3 ⇒ k = - 1 2 . M ≡ 2 , 1 2 , 3 2 Let Q ≡ a , b , c Also, mid-point of R Q is M , a + 2 2 = 2 ⇒ a = 2 b + 3 2 = 1 2 ⇒ b = - 2 c - 1 2 = 3 2 ⇒ c = 4 Hence, Q ≡ 2 , - 2 , 4 . Plane P is perpendicular to line x - 3

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