Set of all points equidistant from (0,b) is a circle. So set of all points at a distance of b from this point is a circle around it touching the x axis (tangent to it as well). At what b will this circle intersect with the parabola?
Solve x^2+(y-b)^2=b^2 with y=x^2/4
y=0, y=2b-4, obviously for the root to make sense it has to have a positive y, since both the circle and parabola lie on y.