Answer
$\displaystyle \frac{x^{2}}{4}+\frac{y^{2}}{8}=1$
Work Step by Step
With the given information,
the foci are $c=2$ units above/below the point (0,0)
the major axis is vertical,
so the standard form the equation is
$\displaystyle \frac{(x-h)^{2}}{b^{2}}+ \displaystyle \frac{(y-k)^{2}}{a^{2}}=1$
The x-intercepts give: $b=2$.
$(h,k)=(0,0)$
$b=2$
$c=2$
From $c^{2}=a^{2}-b^{2}$
$a^{2}=b^{2}+c^{2}=4+4=8$
so the equation is
$\displaystyle \frac{x^{2}}{4}+\frac{y^{2}}{8}=1$
