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Mathematics-Online course: Vector Calculus - Quadratic Curves

Hyperbola

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For the points $ P=(x,y)$ on a hyperbola the difference of the distances from two given points (foci) $ F_{\pm}$ is constant:

$\displaystyle \vert\overrightarrow{PF_-}\vert - \vert\overrightarrow{PF_+}\vert = \pm 2 a $

with $ 2a<\vert\overrightarrow{F_-F_+}\vert$ .

\includegraphics[ width=8.4cm ]{a_hyperbel}

If $ F_{\pm}=(\pm f,0)$ , then we have for the coordinates

$\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\quad b^2 = f^2 - a^2\, , $

and

$\displaystyle r^2 = -\frac{b^2}{1-(f/a)^2\cos^2\theta} $

for the polar coordinates of the points $ P$ . The asymptotes have the slope $ \pm b/a$ .

A parametrisation of the hyperbola branches is given by

$\displaystyle x=\pm a \cosh t,\quad y=b\sinh t $

with $ t\in\mathbb{R}$ .

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  automatically generated 10/30/2007