Precalculus / Unit 9: Conic Sections and Parametric Equations
9.3 Ellipses
Notes
An ellipse is the set of points whose distances to two foci add to a constant.
- Center at the origin: $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$ has a horizontal major axis, vertices $(\pm a, 0)$ and foci $(\pm c, 0)$.
- The larger denominator is $a^2$; its variable tells the direction of the major axis.
- $c^2 = a^2 - b^2$. The length of the major axis is $2a$.
- Eccentricity $\displaystyle e = \frac{c}{a}$ is between 0 and 1.
- Example: $\displaystyle \frac{x^2}{25} + \frac{y^2}{9} = 1$ has $a = 5$, $b = 3$, $c = 4$, foci $(\pm 4, 0)$.
Practice
What are the foci of $\displaystyle \frac{x^2}{25} + \frac{y^2}{9} = 1$?
$c^2 = 25 - 9 = 16$, so $c = 4$ and the major axis is horizontal.
What are the foci of $\displaystyle \frac{x^2}{9} + \frac{y^2}{16} = 1$?
The larger denominator is under $y^2$, so the major axis is vertical, and $c^2 = 16 - 9 = 7$.
What is the center and the length of the major axis of $\displaystyle \frac{(x - 2)^2}{49} + \frac{(y + 1)^2}{4} = 1$?
Center is $(h, k) = (2, -1)$ and $a = 7$, so the major axis has length $2a = 14$.
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