If a circle of constant radius 3k passes through the origin O and meets the coordinate axes at A and B, then the locus of the centroid of triangle OAB is 

Let the centroid of triangle OAB be (p, q).

Hence, points A and B are (3p,0) and (0,3q), respectively.
But diameter of circle, AB = 6k
Hence, $\sqrt{9{\mathrm{p}}^2+9{\mathrm{q}}^2}=6\mathrm{k}$
Therefore, the locus of (p, q) is ${\mathrm{x}}^2+{\mathrm{y}}^2=4{\mathrm{k}}^2$.