College Algebra, Chapter 8, 8.2, Section 8.2, Problem 20

Determine the vertices, foci and eccentricity of the ellipse $x^2 = 4- 2y^2$. Determine the lengths of the major and minor
axes, and sketch the graph.
If we divide both sides by $16$, then we have
$\displaystyle \frac{x^2}{4} + \frac{y^2}{2} = 1$
So, we'll see that the function has the form $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Since the denominator of $x^2$ is larger, then the ellipse
has a horizontal major axis. This gives $\displaystyle a^2 = 4$ and $\displaystyle b^2 = 2$. So,
$\displaystyle c^2 = a^2 - b^2 = 4-2 = 2$. Thus, $\displaystyle a = 2, b = \sqrt{2}$ and
$\displaystyle c = \sqrt{2}$. Then, the following is determined as

$
\begin{equation}
\begin{aligned}
\text{Vertices}& &(\pm a, 0) &\rightarrow (\pm 2, 0)\\
\\
\text{Foci}& &(\pm c, 0) &\rightarrow (\pm \sqrt{2}, 0)\\
\\
\text{Eccentricity (e)}& &\frac{c}{a} &\rightarrow \frac{\sqrt{2}}{2}\\
\\
\text{Length of major axis}& &2a &\rightarrow 4\\
\\
\text{Length of minor axis}& &2b &\rightarrow 2\sqrt{2}
\end{aligned}
\end{equation}
$