The circum center of a triangle lies at the origin and the centroid is midpoint of the line joining the points  $\left(5,5\right)$ and $\left(4,-4\right)$ then the ortho enter lies on the line

The circum center, centroid and ortho center of any triangle are collinear. 

So that the ortho center lies on the line joining the circum center $\left(0,0\right)$ and the centroid $\left(\frac{9}{2},\frac{1}{2}\right)$

Equation of the line joining origin and $\left(\frac{9}{2},\frac{1}{2}\right)$ is $y=\frac{1}{9}x$

Therefore, the equation of required line is  $\boxed{x-9y=0}$