Coordinate Geometry: JEE Main 2026 Mathematics question

Let $A$, $B$ and $C$ be the vertices of a variable right angled triangle inscribed in the parabola $y^2=16x$. Let the vertex $B$ containing the right angle be $(4,8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_0$. Then three times the length of latus rectum of $C_0$ is ______