Four distinct points $(2k, 3k), (1, 0), (0, 1)$and $(0, 0)$ lie on a circle for 

The equation of the circle passing through the point (1, 0), (0, 1) and (0, 0) is $x^2+y^2-x-y=0$

This passes through $(2k ,3k)$ 

$4k^2+9k^2-2k-3k=0\Rightarrow k=0,k=5/13$