Example 6. Try to find the equations of all ellipses with foci at $(\pm 3,0)$, and write down the conditions under which the equation can represent a hyperbola with the same foci as the ellipse.
The foci of the required ellipse lie on the $x$-axis and are symmetric with respect to the coordinate axes, so we can assume its equation to be
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \text {, where } a>b>0 \text {. }
$$

Since the foci are at $(\pm 3,0)$, we have
$$
a^{2}=b^{2}+9, \quad b^{2}=a^{2}-9 \text { . }
$$

The equation of the required ellipse is
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}-9}=1 \text {. }
$$

For $a^{2}-9>0$, we have $a>3$.
Therefore, when $a>3$, the equation represents an ellipse with the same foci as the hyperbola.