check the picture below.
so the ellipse looks more or less like so, let's notice, the distance from the center of the ellipse to a foci is c = 2, and the "a" component or half the major axis is a = 4.
[tex]\bf \textit{ellipse, vertical major axis} \\\\ \cfrac{(x- h)^2}{ b^2}+\cfrac{(y- k)^2}{ a^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h, k\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2- b ^2} \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]
[tex]\bf \begin{cases} h=0\\ k=0\\ a=4\\ c=2 \end{cases}\implies \cfrac{(x-0)^2}{b^2}+\cfrac{(y-0)^2}{4^2}=1 \\\\[-0.35em] ~\dotfill\\\\ c=\sqrt{a^2-b^2}\implies 2=\sqrt{4^2-b^2}\implies 4=16-b^2 \\\\\\ b^2=12\implies b=\sqrt{12} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(x-0)^2}{(\sqrt{12})^2}+\cfrac{(y-0)^2}{4^2}=1\implies \cfrac{x^2}{12}+\cfrac{y^2}{16}=1[/tex]
