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conjugate hyperbola
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(Definition)
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The simplest form of the equation presenting a hyperbola (without the mixed $xy$ -term) in a rectangular coordinate system is got when the coordinate axes coincide with the principal axes of the hyperbola, and it has the form
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(1) |
Here, $a\, (>0)$ is the length of the transverse semiaxis and $b\, (>0)$ the length of the conjugate semiaxis of the hyperbola.
The equation
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(2) |
or $$\frac{x^2}{a^2}-\frac{y^2}{b^2} = -1.$$ presents the 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 $x^2\!+\!y^2 = a^2\!+\!b^2$ .
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"conjugate hyperbola" is owned by pahio. [ full author list (2) ]
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Cross-references: circle, foci, asymptotes, hyperbola, coordinate, rectangular coordinate, equation
There are 2 references to this entry.
This is version 14 of conjugate hyperbola, born on 2004-09-28, modified 2009-03-19.
Object id is 6241, canonical name is ConjugateHyperbola.
Accessed 15075 times total.
Classification:
| AMS MSC: | 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry) |
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Pending Errata and Addenda
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![\begin{pspicture}(-5.5,-4.5)(5.5,4) \psaxes[Dx=10,Dy=10]{->}(0,0)(-4.5,-3.5)(4.5... ...hyperbolas with their common asymptotes\, $y = \pm\frac{b}{a}x$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/6241/js/img3.png "\begin{pspicture}(-5.5,-4.5)(5.5,4) \psaxes[Dx=10,Dy=10]{->}(0,0)(-4.5,-3.5)(4.5... ...hyperbolas with their common asymptotes\, $y = \pm\frac{b}{a}x$} \end{pspicture}")