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JEE Main202311 Apr 2023Evening ShiftMathematicsThree Dimensional GeometryActual

Let P be the plane passing through the points 5 , 3 , 0 , 13 , 3 , - 2 and 1 , 6 , 2 . For α ∈ ℕ , if the distance of the points A 3 , 4 , α and B 2 , α , a from the plane P are 2 and 3 respectively, then the positive value of a is

Options

  1. A6
  2. B3
  3. C5
  4. D4

Correct answer

D. 4

Step-by-step solution

Given, P be the plane passing through the points 5 , 3 , 0 , 13 , 3 , - 2 and 1 , 6 , 2 Now, A B → = 8 i ^ - 2 k ^ , A C → = - 4 i ^ + 3 j ^ + 2 k ^ , So, normal vector to the plane will be, A B → × A C → = i ^ j ^ k ^ 8 0 - 2 - 4 3 2 = 6 i ^ - 8 j ^ + 24 k ^ Now equation of plane will be: 6 x - 8 y + 24 z = d And given plane passes through 5 , 3 , 0 So, 6 × 5 - 8 × 3 + 24 × 0 = d ⇒ d = 6 Hence, equation of plane will be, 6 x - 8 y + 24 z = 6 ⇒ 3 x - 4 y + 12

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