Precalculus / Unit 9: Conic Sections and Parametric Equations
9.4 Hyperbolas
Notes
A hyperbola is the set of points whose distances to two foci differ by a constant.
- $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ opens left and right, with vertices $(\pm a, 0)$ and foci $(\pm c, 0)$, where $c^2 = a^2 + b^2$.
- Asymptotes: $\displaystyle y = \pm\frac{b}{a}x$. For $\displaystyle \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (opens up and down) they are $\displaystyle y = \pm\frac{a}{b}x$.
- The positive term tells you which way it opens.
- Example: $\displaystyle \frac{x^2}{16} - \frac{y^2}{9} = 1$ has $c = 5$ and asymptotes $\displaystyle y = \pm\frac{3}{4}x$.
Practice
What are the foci of $\displaystyle \frac{x^2}{16} - \frac{y^2}{9} = 1$?
$c^2 = 16 + 9 = 25$, so $c = 5$.
What are the asymptotes of $\displaystyle \frac{y^2}{4} - \frac{x^2}{25} = 1$?
Here $a = 2$ (under $y^2$) and $b = 5$, so the slopes are $\displaystyle \pm\frac{a}{b}$.
What type of conic is $4x^2 - 9y^2 = 36$?
Dividing by 36 gives $\displaystyle \frac{x^2}{9} - \frac{y^2}{4} = 1$, which has a minus sign.
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