Rectangular hyperbola with horizontal/vertical asymptotes (Cartesian coordinates)

Rectangular hyperbolas with the coordinate axes parallel to their asymptotes have the equation

$(x-h)(y-k) = m \, \, \,$ .

These are equilateral hyperbolas (eccentricity $\varepsilon = \sqrt 2$ ) with semi-major axis and semi-minor axis given by $\text{[math]}$ .
The simplest example of rectangular hyperbolas occurs when the center (h, k) is at the origin:

$y=\frac{m}{x}\,$

describing quantities x and y that are inversely proportional. By rotating the coordinate axes counterclockwise by 45 degrees, with the new coordinate axes labelled $(x',y')$ the equation of the hyperbola is given by canonical form

$\frac{(x')^2}{(\sqrt{2m})^2}-\frac{(y')^2}{(\sqrt{2m})^2}=1$ .

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Friday 5 October 2012

Rectangular hyperbola with horizontal/vertical asymptotes (Cartesian coordinates)

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