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AP EAMCET202124 Aug 2021Evening ShiftMathematicsThree Dimensional GeometryActual

A variable plane x a + y b + z c =1 , which is at a unit distance from the origin cuts the coordinate axes at A, B and C . If the centroid (x, y, z) of A B C satisfies 1 x^2 + 1 y^2 + 1 z^2 =k , then k equals

Options

  1. A9
  2. B3
  3. C1 9
  4. D1 3

Correct answer

A. 9

Step-by-step solution

x a + y b + z c =1 which is unit distance from the origin cuts the coordinates axes at A, B and C . 1= 1 1 a^2 + 1 b^2 + 1 c^2 1 a^2 + 1 b^2 + 1 c^2 =1 Coordinate of vertices of A B C is A(a, 0,0), B(0, b, 0), C(0,0, c) Centroid of A B C is ( a 3 , b 3 , c 3 ) Now, ( a 3 , b 3 , c 3 ) satisfied the equation 1 x^2 + 1 y^2 + 1 z^2 =k 9 a^2 + 9 b^2 + 9 c^2 =k9 ( 1 a^2 + 1 b^2 + 1 c^2 )=k [ 1 a^2 + 1 b^2 + 1 c^2 =1 ] k=9

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