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JEE Main202120 Jul 2021Evening ShiftMathematicsThree Dimensional GeometryActual

The lines x = a y - 1 = z - 2 and x = 3 y - 2 = b z - 2 , ( a b ≠ 0 ) are coplanar, if:

Options

  1. Ab = 1 , a ∈ R - 0
  2. Ba = 1 ,   b ∈ R - 0
  3. Ca = 2 ,   b = 2
  4. Da = 2 ,   b = 3

Correct answer

A. b = 1 , a ∈ R - 0

Step-by-step solution

Given lines can be written as x + 1 a = y - 0 1 = z - 1 a x + 2 3 = y - 0 1 = z - 0 3 b We know that lines x - x 1 a 1 = y - y 1 b 1 = z - z 1 c 1 and x - x 2 a 2 = y - y 2 b 2 = z - z 2 c 2 are coplanar if x 2 - x 1 y 2 - y 1 z 2 - z 1 a 1 b 1 c 1 a 2 b 2 c 2 = 0 . Hence, - 1 0 - 1 a 1 a 3 1 3 b = 0 where, b ≠ 0 ⇒ a ≠ 0 , since a b ≠ 0 . ⇒ - 3   b - a - 1 a - 3 = 0 ⇒ a - 3 b - a + 3 = 0 ⇒ b = 1 ∴ b = 1 , a ∈ R - 0

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