Rectangular Hyperbola
Rectangular or equilateral hyperbolas have equal semiaxes (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.
