# Hyperbola

The hyperbola is the locus of points on the plane whose difference of distances to two fixed points, foci, are constant.

### Elements of the Hyperbola

**Foci**

The foci are the fixed points of the hyperbola. They are denoted by **F ** and **F'**.

**Transverse Axis** or real axis

The tranverse axis is the line segment between the foci.

**Conjugate Axis** or imaginary axis

The conjuagate axis is the perpendicular bisector of the line segment (transverse axis).

**Center **

The center is the point of intersection of the axes and is also the center of symmetry of the hyperbola.

**Vertices**

The points A and A' are the points of intersection of the hyperbola with the transverse axis.

**Focal Radii**

The focal radii are the line segments that join a point on the hyperbola with the foci: **PF** and **PF'**.

**Focal Length**

The focal length is the line segment , which has a length of **2c**.

**Semi-Major Axis**

The semi-major axis is the line segment that runs from the center to a vertex of the hyperbola. Its length is **a**.

**Semi-Minor Axis**

The semi-minor axis is a line segment which is perpendicular to the semi-major axis. Its length is **b**.

**Axes of Symmetry**

The axes of symmetry are the lines that coincide with the transversal and conjugate axis.

**Asymptotes**

The asymptotes are the lines with the equations:

**The Relationship Between the Semiaxes**:

**Eccentricity of a Hyperbola**:

## Equation of the Hyperbola

### Hyperbolas Centered at (0, 0)

#### Horizontal Transverse Axis

**F'(−c, 0) and F(c, 0)**

#### Vertical Transverse Axis

**F'(0, −c) and F(0, c)**

### Hyperbolas Centered at (x_{0}, y_{0})

#### Horizontal Transverse Axis

**F(x _{0}+c, y_{0}) and F'(x_{0}− c, y_{0})**

By removing the denominators, an equation is obtained in the form:

Keep in mind that A and B have opposite signs.

#### Vertical Transverse Axis

**F(x _{0}, y_{0}+c) and F'(x_{0}, y_{0}−c)**

## Rectangular Hyperbola

Hyperbolas where the semiaxes are equal are called rectangular or equilateral hyperbolas (a = b).

The equation of a rectangular hyperbola is:

The equations of the asymptotes are:

,

That is, the angle bisectors of the quadrants.

The eccentricity is:

### Equation of a Rectangular Hyperbola

To switch the asymptotes to those determined by the x and y-axis, turn the asymptote −45° about the origin.

If it is rotated 45°, the hyperbola is in the second and fourth quadrant.