Hi,
I can show you two ways to approach this problem.
 First approach:
 The equation of the circle with center $(h, k)$ and radius $r$ is $(x  h)^2 + (y  k)^2 = r^{2}.$ Use this fact and the fact that the circle passes through $(0,0), (1,1)$ and $(6,0)$ to write three equations in $h, k$ and $r.$ A careful look at these equations shows that two of them can be easily solved for $h.$ Once you know $h$ you can solve the equations for $k$ and $r.$
 Second approach:
 The slope of the line $3x + 2y = 0$ is $\large \frac{3}{2}$ and hence the line trough $(0,0)$ and perpendicular to $3x + 2y = 0$ has slope $\large \frac{2}{3}.$ Write the equation of this line.The center of the circle lies on this line and this fact gives you an equation relating $h$ and $k$ directly. Use the equations developed in the first approach to again solve for $h.$ This and the new equation you just found allow you to find $k.$
I hope this helps,
Penny
