Six points (x_i, y_i ), i=1,2, , 6 are taken on the circle x^2+y^2=4 such that _ i=1 ^6 x_i=8 and _ i=1 ^6 y_i=4 . The line segment joining orthocentre of a triangle made by any three points and the centroid of the triangle made by other three points passes through a fixed points ( h , k ) , then h + k is

Step-by-step solution

Let _ i=1 ^6 x_i= and _ i=1 ^6 y_i= Let O be the orthocentre of the triangle made by ( . x ₁, y ₁ ) , ( x ₂, y ₂ ) and ( . x ₃, y ₃ ) O is (x₁+x₂+x₃ y₁+y₂+y₃ ) ( ₁ ₁ ) Similarly let G be the centroid of the triangle made by other three points G is ( x₄+x₅+x₆ 3 , y₄+y₅+y₆ 3 ) G is ( - ₁ 3 , ₁- ₁ 3 ) The point dividing O G in the ratio 3: 1 is ( 4 , 4 ) (2,1) h+k=3