10. (20 points) Let the center of a moving circle be on the parabola $y=\frac{1}{4} x^{2}$, and the radius of the circle be equal to the y-coordinate of the center of the circle. Find the set of all points on such circles.

保留源文本的换行和格式，翻译结果如下：

10. (20 points) Let the center of a moving circle be on the parabola $y=\frac{1}{4} x^{2}$, and the radius of the circle be equal to the y-coordinate of the center of the circle. Find the set of all points on such circles.
10. Let the center of a moving circle be \( C\left(a, \frac{1}{4} a^{2}\right) \), and \( P(x, y) \) be any point on \( \odot C \). Then
\[
(x-a)^{2}+\left(y-\frac{1}{4} a^{2}\right)^{2}=\left(\frac{1}{4} a^{2}\right)^{2},
\]
which simplifies to \((2-y) a^{2}-4 x a+2\left(x^{2}+y^{2}\right)=0\).
When \( y=2 \), \( a=\frac{x^{2}+4}{2 x} \), so \( x \neq 0 \).
When \( y \neq 2 \), since \( a \in \mathbf{R} \), we have
\[
\Delta=16 x^{2}-8(2-y)\left(x^{2}+y^{2}\right) \geqslant 0.
\]
Simplifying, we get \( y\left(x^{2}+y^{2}-2 y\right) \geqslant 0 \).
Thus, \(\left\{\begin{array}{l}y \geqslant 0, \\ x^{2}+y^{2}-2 \geqslant 0\end{array}\right.\) or \(\left\{\begin{array}{l}y \leqslant 0, \\ x^{2}+y^{2}-2 y \leqslant 0 .\end{array}\right.\) Therefore, \(\left\{\begin{array}{l}y \leqslant 0, \\ x^{2}+(y-1)^{2} \geqslant 1\end{array}\right.\) or \(x=y=0\). Hence, the required set is
\[
\left\{(x, y) \mid x^{2}+(y-1)^{2} \geqslant 1, y \geqslant 0\right\} \backslash\{(0,2)\}.
\]