Conjugate hyperbola

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In [[geometry]], a '''conjugate hyperbola''' to a given [[hyperbola]] shares the same [[asymptote]]s but lies in the opposite two sectors of the plane compared to the original hyperbola.	In [[geometry]], a '''conjugate hyperbola''' to a given [[hyperbola]] shares the same [[asymptote]]s but lies in the opposite two sectors of the plane compared to the original hyperbola.
A hyperbola and its conjugate may be constructed as [[conic section]]s derived from an intersecting plane and cutting tangent double cones sharing the same [[apex (geometry)|apex]].	A hyperbola and its conjugate may be constructed as [[conic section]]s derived from parallel intersecting planes and cutting tangent double cones sharing the same [[apex (geometry)|apex]].
Using [[analytic geometry]], the hyperbolas satisfy the symmetric equations
==Construction==
:<math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1</math> with vertices (''a'',0) and (–''a'',0), and
Two [[cone]]s with common apex at the origin are tangent along the lines ''y'' = ''mx'' and ''y'' = –''mx''. The cone around the x-axis is
:<math>y^2 + z^2 = (mx)^2</math> and the one about the y-axis is	:<math>\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1</math> with vertices (0,''b'') and (0,–''b'').
In case ''a'' = ''b'' they are rectangular hyperbolas, and a reflection of the plane in an asymptote exchanges the conjugates.
:<math>x^2 + z^2 = (y/m)^2 .</math>
The intersections with plane ''z'' = ''b'' yield two hyperbolas:
:<math>b^2 = (mx)^2 - y^2 \ \ \text{and} \ \ b^2 = (y/m)^2 - x^2 ,</math> or equivalently
:<math>1 = \frac{(mx)^2}{b^2} - \frac{y^2}{b^2} \ \ \text{and} \ \ 1 = \frac{y^2}{(mb)^2} - \frac{x^2}{b^2}.</math>
These hyperbolas are conjugate with respect to each other.
The rectangular case is recovered with ''m'' = 1 = ''b''.
==History==	==History==

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