I used distance formula and differentiated it wrt to x
but I'm not getting the answer.
could you please do this using that and tell where I'm going wrong?
please!

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

4y+(y-b)^2=b^2

y^2+y(4-2b)=0

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.