College Algebra, Chapter 8, 8.2, Section 8.2, Problem 12

Determine the vertices, foci and eccentricity of the ellipse $4x^2 + 25y^2 = 100$. Determine the lengths of the major and minor
axes, and sketch the graph.
If we divide both sides by $100$, then we have
$\displaystyle \frac{x^2}{25} + \frac{y^2}{4} = 1$
Since the denominator of $x^2$ is larger, then the ellipse has a horizontal major axis. This gives $a^2 = 25$ and $b^2 = 4$, so
$c^2 = a^2 - b^2 = 25 - 4 = 21$. Thus, $a=5$, $b =2$ and $c = \sqrt{21}$. Then, the following are determined as

$
\begin{equation}
\begin{aligned}
\text{Vertices}& &(\pm a, 0) &\rightarrow (\pm 5, 0)\\
\\
\text{Foci}& &(\pm c, 0) &\rightarrow (\pm\sqrt{21}, 0)\\
\\
\text{Eccentricity (e)}& &\frac{c}{a} &\rightarrow \frac{\sqrt{21}}{5}\\
\\
\text{Length of major axis}& &2a &\rightarrow 10\\
\\
\text{Length of minor axis}& &2b &\rightarrow 4
\end{aligned}
\end{equation}
$