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NTA Abhyas JEE Main2020MathematicsThree Dimensional GeometryPractice

The plane x 1 + y 2 + z 3 = 1 intersect x -axis, y -axis and z -axis at A , B and C respectively. If the distance between the origin and the centroid of ∆ A B C is k 1 units and the volume of the tetrahedron O A B C is k 2 cubic units, then the value of k 1 2 k 2 is equal to (where O is the origin)

Options

  1. A21
  2. B14 9
  3. C63
  4. D14 3

Correct answer

B. 14 9

Step-by-step solution

x 1 + y 2 + z 3 = 1 cuts x -axis, y -axis, z -axis at A 1 , 0 , 0 , B 0 , 2 , 0 , C 0 , 0 , 3 respectively Centroid of ∆ A B C is 1 3 , 2 3 , 1 Distance between centroid and origin is k 1 = 1 9 + 4 9 + 1 ⇒ k 1 = 14 3 units Volume of tetrahedron O A B C = 1 6 × O A × O B × O C k 2 = 1 6 × 1 × 2 × 3 = 1 cubic unit k 1 2 k 2 = 14 9 1 = 14 9

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