The center of the circle is approximately equal to \((0,0.542)\) and radius of the circle is approximately equal to 0.4571.
According to the definition, the curve \(y=2x^2\) is the geometric location of the points that are of the same distance from the focus with coordinates \(F: (x=0, y=1/8)\) and the directrix with the relation \(y=-1/8\). Therefore, in the diagram below, \(\bar{FP}=\bar{AP}\) and their adjacent angles in the formed triangle are equal.

Also, by the symmetry of parabola, the segments \(BP\) and \(FP\) make an equal angle with the segment perpendicular to the parabola at the point \(P\), and according to the property of the circle, the segment perpendicular to the point \(P\) is the radius of the circle.

Thus the angles marked in the figure are equal, and \(CFAP\) is a parallelogram, and therefore \(\bar{CF}=\bar{AP}\), from which we can conclude \(C_y=P_y+1/4\).

\[\begin{aligned} CF = AP \implies& \\ C_y - F_y = P_y - A_y \implies& \\ C_y - \frac{1}{8} = P_y + \frac{1}{8} \implies& C_y=P_y+\frac{1}{4} \end{aligned}\]

On the other hand, the radius of the circle is equal to \(1-C_y\). Therefore \(\bar{CP}=1-C_y\). If we put this next to \(C_y=P_y+1/4\), we can determine \(C_y\) and \(R\).

\[\begin{aligned} CM = CP \implies& (1 - C_y)^2 = (C_y - P_y)^2 + (0 - P_x)^2 \\ \implies& (1 - C_y)^2 = \frac{1}{4}^2 + \frac{P_y}{2} \\ \implies& 1 - 2C_y + C_y^2 = \frac{1}{4}^2 + \frac{C_y-1/4}{2} \\ \implies& C_y^2 - \frac{5}{2}C_y + \frac{17}{16} = 0 \\ \implies& C_y = \frac{5}{4} - \frac{1}{\sqrt{2}}, R = \frac{1}{\sqrt{2}} - \frac{1}{4} \end{aligned}\]