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JEE MainMathematicsStraight Lines

Let the circumcentre of a variable triangle be at the origin. If the locus of its centroid is the circle x^2 + y^2 - 2x - 4y + 4 = 0 , then the locus of its orthocentre is also a circle. The sum of the coordinates of the centre and the radius of this circle is

Options

  1. A6
  2. B2
  3. C12
  4. D4

Correct answer

C. 12

Step-by-step solution

Let the circumcentre be O(0,0) , the centroid be G(h,k) and the orthocentre be H(x,y) . We know that in any triangle, the centroid G divides the line segment joining the circumcentre O and the orthocentre H internally in the ratio 1:2 . Using the section formula, the coordinates of G are given by: h = 1 x + 2 0 1+2 = x 3 k = 1 y + 2 0 1+2 = y 3 It is given that the locus of the centroid G(h,k) is the circle h^2 + k^2 - 2h - 4k + 4 = 0 . Substituting h = x 3 and k = y 3 into the equation of the circle: ( x 3 )^2 + (

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