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JEE Main202330 Jan 2023Evening ShiftMathematicsThree Dimensional GeometryActual

If a plane passes through the points ( - 1 , k , 0 ) , ( 2 , k , - 1 ) , ( 1 , 1 , 2 ) and is parallel to the line x - 1 1 = 2 y + 1 2 = z + 1 - 1 , then the value of k 2 + 1 ( k - 1 ) ( k - 2 ) is

Options

  1. A17 5
  2. B5 17
  3. C6 13
  4. D13 6

Correct answer

D. 13 6

Step-by-step solution

Given, A plane passes through the points ( - 1 , k , 0 ) , ( 2 , k , - 1 ) , ( 1 , 1 , 2 ) and is parallel to the line x - 1 1 = 2 y + 1 2 = z + 1 - 1 , So, on rearranging x - 1 1 = 2 y + 1 2 = z + 1 - 1 we get, ⇒ x - 1 1 = y + 1 2 1 = z + 1 - 1 Now given point : A ( - 1 , k , 0 ) , B ( 2 , k , - 1 ) , C ( 1 , 1 , 2 ) So, C A → = - 2 i ^ + ( k - 1 ) j ^ - 2 k ^ , C B → = i ^ + ( k - 1 ) j ^ - 3 k ^ And C A → × C B → = i ^ j ^ k ^ - 2 k - 1 - 2 1 k - 1 - 3 ⇒ C A → &#215

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