check the picture below.
notice, the ellipse is taller than it's wider, meaning the major axis is over the vertical, meaning the "a" component will go with the fraction with the "y" variable,
[tex]\bf \textit{ellipse, vertical major axis}
\\\\
\cfrac{(x- h)^2}{ b^2}+\cfrac{(y- k)^2}{ a^2}=1\qquad
\begin{cases}
h=0\\
k=0\\
b=\frac{1}{2}\\\\
a=\frac{3}{2}
\end{cases}
\\\\\\
\cfrac{(x- 0)^2}{ \left( \frac{1}{2} \right)^2}+\cfrac{(y- 0)^2}{ \left( \frac{3}{2} \right)^2}=1\implies \cfrac{x^2}{\frac{1}{4}}+\cfrac{y^2}{\frac{9}{4}}=1\implies \cfrac{4x^2}{1}+\cfrac{4y^2}{9}=1
\\\\\\
4x^2+\cfrac{4y^2}{9}=1[/tex]
