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JEE Main2017MathematicsThree Dimensional GeometryActual

If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A , B & C , then the locus of the centroid of Δ A B C is

Options

  1. A1 x 2 + 1 y 2 + 1 z 2 = 1
  2. B1 x 2 + 1 y 2 + 1 z 2 = 3
  3. C1 x 2 + 1 y 2 + 1 z 2 = 9
  4. D1 x 2 + 1 y 2 + 1 z 2 = 1 9

Correct answer

A. 1 x 2 + 1 y 2 + 1 z 2 = 1

Step-by-step solution

Let centroid be h ,   k ,   l x -intercept = 3 h   y -intercept = 3 k and z - intercept = 3 l Now, the equation of plane is x 3 h + y 3 k + z 3 l = 1 . Distance from 0 ,   0 ,   0 ⇒ 1 1 9 h 2 + 1 9 k 2 + 1 9 l 2 = 3 ⇒ 1 x 2 + 1 y 2 + 1 z 2 = 1

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