3. Given the ellipse $x^{2}+4(y-a)^{2}=4$ and the parabola $x^{2}=2 y$ have common points. Then the range of values for $a$ is $\qquad$ .
3. $\left[-1, \frac{17}{8}\right]$.

Notice that, the parametric equation of the ellipse is
$$
\left\{\begin{array}{l}
x=2 \cos \theta, \\
y=a+\sin \theta
\end{array}(\theta \in \mathbf{R})\right. \text {. }
$$

Substituting into the parabola equation, we get
$$
\begin{array}{l}
4 \cos ^{2} \theta=2(a+\sin \theta) \\
\Rightarrow a=2 \cos ^{2} \theta-\sin \theta \\
=-2\left(\sin \theta+\frac{1}{4}\right)^{2}+\frac{17}{8} .
\end{array}
$$

Since $\sin \theta \in[-1,1]$, then $a \in\left[-1, \frac{17}{8}\right]$.