Answer
$\frac{y^2}{2}-\frac{x^2}{2}=1$
See graph.
Work Step by Step
1. Given foci $(0,-2)$ and $(0,2)$, asymptote $y=-x$, we have a vertical transverse axis, center $(0,0)$ (midpoint of foci), $c=2, \frac{b}{a}=1$, $a^2+b^2=c^2=4$,
2. Thus $a^2=b^2=2$ and the equation is $-\frac{x^2}{2}+\frac{y^2}{2}=1$ or $\frac{y^2}{2}-\frac{x^2}{2}=1$
3. See graph.
