M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesLearning module LM 10.1: Parametrized Curves:Learning module LM 10.2: Calculus with Parametrized Curves:Learning module LM 10.3: Polar Coordinates:Learning module LM 10.4: Areas and Lengths of Polar Curves:Learning module LM 10.5: Conic Sections:Slicing a coneEllipses Hyperbolas Parabolas and directrices Completing the square Learning module LM 10.6: Conic Sections in Polar Coordinates:Chapter 12: Vectors and the Geometry of SpaceChapter 13: Vector FunctionsChapter 14: Partial DerivativesChapter 15: Multiple Integrals |
HyperbolasThe equations for hyperbolas are very similar to those for ellipses, only with a minus sign.
Hyperbolas can also be described by a distance formula.
A hyperbola consists of two pieces, one with $L_1-L_2 = 2a$ and one with $L_1 - L_2 = -2a$. The four ends approach the lines $y = \pm \frac{b}{a}x$ (for the first kind of hyperbola) or $y = \pm \frac{a}{b}x$ (for the second kind). These lines are called asymptotes. Example: The asymptotes of the hyperbola $x^2 - y^2 = 2$ are the diagonal lines $y = \pm x$. If you rotate this hyperbola counter-clockwise by 45 degrees, you get the curve $y = \frac{1}{x}$, whose asymptotes are the $x$ and $y$ axes. |