Hyperbola

Definition and Properties of a Hyperbola

A hyperbola is a plane curve such that the difference of the distances from any point of the curve to two other fixed points (called the foci of the hyperbola) is constant. The midpoint of the line segment joining the foci is called the center of the hyperbola. The distance from the center to a focus is called the focal distance and denoted by c.

Any hyperbola consists of two distinct branches. The points on the two branches that are closest to each other are called the vertices. The distance of a vertex to the center is denoted by a.

The line passing through the vertices of a hyperbola is called the transverse axis. (It contains the segment of length 2a between the vertices). The transverse axis of a hyperbola is its line of symmetry. Another line of symmetry is perpendicular to the transverse axis and is called the conjugate axis. (It contains the segment of length 2b with midpoint at the center of the hyperbola).

The standard (or canonical) equation of the hyperbola is

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

The absolute value of the difference of the distances from any point of a hyperbola to its foci is constant:

\[\left|{r_1 - r_2}\right| = 2a\]

where \(r_1,\) \(r_2\) are the distances from an arbitrary point \(P\left({x,y}\right)\) of the hyperbola to the foci \(F_1\) and \(F_2,\) \(a\) is the transverse semi-axis of the hyperbola.

The distances from any point of a hyperbola to its foci — Figure 2.

Equations of the Asymptotes of a Hyperbola

\[y = \pm \frac{b}{a}x\]

Relationship between Semi-Axes of a Hyperbola and its Focal Distance

\[c^2 = a^2 + b^2\]

where \(c\) is the focal distance, \(a\) is the transverse semi-axis of the hyperbola, \(b\) is the conjugate semi-axis.

Eccentricity of a Hyperbola

\[e = \frac{c}{a} \gt 1\]

Equations of the Directrices of a Hyperbola

The directrix of a hyperbola is a straight line perpendicular to the transverse axis of the hyperbola and intersecting it at the distance \({\frac{a}{e}}\) from the center. A hyperbola has two directrices spaced on opposite sides of the center. The equations of the directrices are given by

\[x = \pm \frac{a}{e} = \pm \frac{a^2}{c}\]

Parametric Equations of a Hyperbola (Right Branch)

\[\left\{ \begin{aligned} x &= a\cosh t \\ y &= b\sinh t \end{aligned} \right.,\;\; 0 \le t \le 2\pi\]

where \(a, b\) are the semi-axes of the hyperbola, \(t\) is a parameter.

General Equation of a Hyperbola

\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]

where \(B^2 - 4AC \gt 0.\)

General Equation of a Hyperbola with Axes Parallel to the Coordinate Axes

\[Ax^2 + Cy^2 + Dx + Ey + F = 0\]

where \(AC \lt 0.\)

Precalculus

Analytic Geometry