Hyperbolas

The definition

An ellipse uses a sum, a hyperbola uses a difference. The result looks very different: a hyperbola has two separate branches that open away from each other.

Standard equations

The segment between the two vertices is the transverse axis, of length \(2a\). The foci are at distance \(c\) from the center, with

Here \(a^2\) is always under the positive term, and that term tells you the direction. Unlike the ellipse, \(a\) doesn't have to be larger than \(b\).

Asymptotes

The branches get closer and closer to two lines, the asymptotes, without ever touching them. To sketch them, draw the box with sides \(2a\) and \(2b\) centered at the center; the asymptotes are its extended diagonals.

Eccentricity and rectangular hyperbolas

The eccentricity is again \(e = \frac{c}{a}\), and since \(c > a\) for a hyperbola, \(e > 1\).

When \(a = b\), the asymptotes are \(y = \pm x\), which are perpendicular. Such a hyperbola is called rectangular. Rotated by \(45°\), a rectangular hyperbola has the coordinate axes as asymptotes and equation \(xy = k\). That's the graph of inverse variation, like \(y = \frac{12}{x}\).

Hyperbolas centered at \((h, k)\)

Practice