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'conjugate hyperbola'
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| Title of object: |
conjugate hyperbola |
| Canonical Name: |
ConjugateHyperbola |
| Type: |
Definition |
| Created on: |
2004-09-28 07:19:07 |
| Modified on: |
2006-02-14 07:16:05 |
| Classification: |
msc:51N20 |
| Defines: |
transverse axis, conjugate axis, mixed term |
Revision comment (for changes between this and next version):
| page images rendering fixed |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx} |
Content:
The simplest form of the equation presenting a hyperbola (without the \PMlinkescapetext{{\em mixed $xy$-term}}) in a rectangular coordinate system is got when the coordinate axes coincide with the \PMlinkescapetext{principal} axes of the hyperbola, and it has the form
\begin{align}
\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1.
\end{align}
Here, $a \,(>0)$ is the \PMlinkescapetext{length of the {\em transverse semiaxis}} and $b \,(>0)$ the \PMlinkescapetext{length of the {\em conjugate semiaxis}} of the hyperbola.
The equation
\begin{align}
\frac{y^2}{b^2}-\frac{x^2}{a^2} = 1
\end{align}
or
$$\frac{x^2}{a^2}-\frac{y^2}{b^2} = -1.$$
presents the {\em conjugate hyperbola} of (1).\, Its transverse axis is the conjugate axis of (1) and its conjugate axis the transverse axis of (1).\, Both hyperbolas are conjugate hyperbolas of each other.\, They have the common asymptotes
$$\frac{x^2}{a^2}-\frac{y^2}{b^2} = 0$$
and their foci are on the circle \,$x^2+y^2 = a^2+b^2$.
\begin{center}
\includegraphics{conjughyperb}
\end{center} |
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