If the circumcentre of a triangle lies at the origin and the centroid is the middle point of the line joining the point  is a2+1, a2+1 and (2a,−2a) then the orthocentre satisfies the equation

The circumcentre is at the origin and the centroidis at a2+1+2a2,a2+1−2a2 i.e at (a+1)22,(a−1)22Let (x, y) be the coordinates of the orthocentre. Since centroid divides the line segment joining circumcentre and orthocentre in the ratio 1 : 2. ∴ (a+1)22=x3 and a(a−1)22=y3⇒ (a+1)2(a−1)2=xy⇒(a−1)2x=(a+1)2y