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AP EAMCET2012MathematicsThree Dimensional Geometry

The equation of the sphere through the points (1,0,0),(0,1,0) and (1,1,1) and having the smallest radius

Options

  1. A3 (x^2+y^2+z^2 )-4 x-4 y-2 z+1=0
  2. B2 (x^2+y^2+z^2 )-3 x-3 y-z+1=0
  3. Cx^2+y^2+z^2-x-y+z+1=0
  4. Dx^2+y^2+z^2-2 x-2 y+4 z+1=0

Correct answer

A. 3 (x^2+y^2+z^2 )-4 x-4 y-2 z+1=0

Step-by-step solution

Given points are A(1,0,0), B(0,1,0) and C(1,1,1) aligned & A B= (0-1)^2+(1-0)^2+0^2 = 2 & B C= (0-1)^2+0^2+1^2- 2 & C A= 0^2+1^2+1^2 = 2 & aligned M A C is an equilateral triangle. Centre of sphere = Centroid of A B C=C^ ( 2 3 , 2 3 , 1 3 ) Radius of sphere A C^ aligned & = ( 2 3 -1 )^2+ ( 2 3 )^2+ ( 1 3 )^2 & = 1 9 + 4 9 + 1 9 = 1 3 6 aligned Equation of sphere is gathered (x- 2 3 )^2+ (y- 2 3 )^2+ (z- 1 3 )^2= ( 6 3 )^2 x^2+y^2+z^2- 4 x 3 - 4 y 3 - 2 z 3 + 4 9 + 4 9 + 1 9 = 6 9 = 2 3 3 (x^2+y^2+z^2 )-4 x-4 y-2 z+

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