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KCET2021MathematicsThree Dimensional Geometry

If a plane meets the coordinate axes at A, B and C in such a way that the centroid of A B C is at the point (1,2,3) , then the equation of the plane is

Options

  1. Ax 1 + y 2 + z 3 =1
  2. Bx 3 + y 6 + z 9 =1
  3. Cx 1 + y 2 + z 3 = 1 3
  4. Dx 1 - y 2 + z 3 =-1

Correct answer

B. x 3 + y 6 + z 9 =1

Step-by-step solution

Given, plane meets the coordinate axes at A, B and C . Let the plane meets the coordinate axes at the points A(a, 0,0), B(0, b, 0), C(0,0, c) The equation of plane is x a + y b + z c =1...(i) Centroid formula says, aligned &x= x₁+x₂+x₃ 3 , y= y₁+y₂+y₃ 3 &z= z₁+z₂+z₃ 3 aligned Centroid of A B C= ( a 3 , b 3 , c 3 ) Given centroid =(1,2,3) ( a 3 , b 3 , c 3 ) Therefore, a=3, b=6 and c=9 Put the value of a, b and c in equation of plane (i), we get x 3 + y 6 + z 9 =1

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