# 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.

#### Example

Calculate the vertices and foci of a rectangular hyperbola of equation .

The coordinates of the vertices are on the bisector of the first and third quadrant and the first and second coordinate coincide, that is to say, x = y. Also, Point A belongs to the curve of the hyperbola.

The length of the semi-axis, a, is the distance from the origin to Vertex A.

The length of the semi-axis, c, is the distance from the origin to Point C.