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AP EAMCET2001MathematicsThree Dimensional Geometry

A variable plane is at a constant distance h from the origin and meets the coordinate axes in A, B, C . Locus of centroid of A B C is

Options

  1. Ax^2+y^2+z^2=h⁻²
  2. Bx^2+y^2+z^2=4 h⁻²
  3. Cx^2+y^2+z^2=16 h^2
  4. D1 x^2 + 1 y^2 + 1 z^2 = 9 h^2

Correct answer

D. 1 x^2 + 1 y^2 + 1 z^2 = 9 h^2

Step-by-step solution

The centroid of the triangle is C ( a 3 , b 3 , c 3 ) The direction coines of the line O C are ( h a , h b , h c ) . As we know that, array rlrl & l^2+m^2+n^2 & =1 & & h^2 a^2 + h^2 b^2 + h^2 c^2 & =1 & & 1 a^2 + 1 b^2 + 1 c^2 & = 1 h^2 & & ( 3 a )^2+ ( 3 b )^2+ ( 3 c )^2 & = 9 h^2 array The locus of the centroid is 1 x^2 + 1 y^2 + 1 z^2 = 9 h^2

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