3. The equation of a circle whose center is on the parabola $x^{2}=2 y$ and which is tangent to the directrix of the parabola and the $y$-axis is
荅蛙 $(x+1)^{2}+\left(y-\frac{1}{2}\right)^{2}=1$ and $(x-1)^{2}+\left(y-\frac{1}{2}\right)^{2}=1$.
Analysis The directrix equation of the parabola $x^{2}=2 y$ is $y=-\frac{1}{2}$.
Let the center of the required circle be $\left(x_{0}, y_{0}\right)$, then $x_{0}^{2}=2 y_{0}$, and $\left|x_{0}\right|=y_{0}+\frac{1}{2}$, solving this gives $x_{0}= \pm 1, y_{0}=\frac{1}{2}$. Therefore, the equation of the required circle is $(x \pm 1)^{2}+\left(y-\frac{1}{2}\right)^{2}=1$.