Hyperbolas

A hyperbola is the set of points such the difference of the distances between a point and two fixed points, the foci, is constant and less than the distance between the foci.

A standard hyperbola centered at the origin with foci on the $x$-axis, is

\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 .\]

This hyperbola will have

• foci at $(\pm \sqrt{a^2 + b^2}, 0)$, (proof (mathforum.org)),
• assymptotes along $y = \pm\frac{b}{a}x$,
• and vertices at $(\pm a, 0)$.