jmc

Note that the lines $4x-3y=30$ and $4x-3y=-10$ are parallel, so the center of the circle lies on the line which is exactly halfway between these lines, which is $4x-3y=10.$



Solving the system $2x+y=0$ and $4x-3y=10,$ we find $x=1$ and $y=-2.$ Therefore, the center of the circle is $\boxed{(1,-2)}.$