Answer
$\displaystyle \frac{y^{2}}{4}-\frac{x^{2}}{16}=1$
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 vertices, $\quad \Rightarrow\quad a^{2}=4$
Given the asymptotes, $\quad \displaystyle \Rightarrow\quad \frac{a^{2}}{b^{2}}=\frac{1}{4}\quad \Rightarrow\quad 4a^{2}=b^{2}.$
$\quad \Rightarrow\quad b^{2}=16$
The equation is $\displaystyle \frac{y^{2}}{4}-\frac{x^{2}}{16}=1$
