College Algebra, Chapter 8, Review Exercises, Section Review Exercises, Problem 12

Determine the center, vertices, foci, eccentricity and lengths of the major and minor axes of the ellipse $\displaystyle 9x^2 + 4y^2 = 1$. Then sketch its graph.

If we rewrite the equation as $\displaystyle \frac{x^2}{\displaystyle \frac{1}{9}} + \frac{y^2}{\displaystyle \frac{1}{4}} = 1$, we'll see that the equation now has the form $\displaystyle \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$. Since the denominator of $y^2$ is larger, the ellipse has a vertical major axis. This gives $\displaystyle a^2 = \frac{1}{4}$ and $\displaystyle b^2 = \frac{1}{9}$, so $\displaystyle c^2 = a^2 - b^2 = \frac{1}{4} - \frac{1}{9} = \frac{5}{36}$. Thus, $\displaystyle a = \frac{1}{2}, b = \frac{1}{3}$ and $\displaystyle c = \frac{\sqrt{5}}{6}$. Then, the following are determined as

vertices $\displaystyle (0,\pm a) \to \left(0, \pm \frac{1}{2}\right) $

foci $\displaystyle (0,\pm c) \to \left( 0, \pm \frac{\sqrt{5}}{6} \right)$

eccentricity $\displaystyle \frac{c}{a} \to \frac{\sqrt{5}}{3}$

length of the major axis $2a \to 1$

length of the minor axis $\displaystyle 2b \to \frac{2}{3}$

Therefore, the graph is

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