Answer
$y^{2}-x^{2}=1$
Work Step by Step
Foci are on the y-axis: $\quad \displaystyle \frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$
Center-to-focus distance: $\quad c=\sqrt{a^{2}+b^{2}}$
Foci: $\quad (0, \pm c)$
Vertices: $\quad (0, \pm a)$
Asymptotes: $\quad y=\displaystyle \pm\frac{a}{b}x$
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Given the foci, $ c^{2}=2\quad \Rightarrow\quad a^{2}+b^{2}=2$
Given the asymptotes, $\quad \displaystyle \Rightarrow\quad\frac{a}{b}=1\quad \Rightarrow\quad a=b.$
Substituting into $a^{2}+b^{2}=2$,
$2a^{2}=2$,
$a^{2}=1\quad(b^{2}=1)$
The equation is $\displaystyle \frac{y^{2}}{1}-\frac{x^{2}}{1}=1$
or
$y^{2}-x^{2}=1$
