For example, the eccentricity of a rectangular hyperbola (θ = 45°, a = b ) equals the square root of two: e =

Every hyperbola has a conjugate hyperbola, in which the transverse and conjugate axes are exchanged without changing the asymptotes. The equation of the conjugate hyperbola of

If the graph of the conjugate hyperbola is rotated 90° to restore the east-west opening orientation (so that x becomes y and vice versa), the equation of the resulting rotated conjugate hyperbola is the same as the equation of the original hyperbola except with a and b exchanged. For example, the angle θ of the conjugate hyperbola equals 90° minus the angle of the original hyperbola. Thus, the angles in the original and conjugate hyperbolas are complementary angles, which implies that they have different eccentricities unless θ = 45° (a rectangular hyperbola). Hence, the conjugate hyperbola does not in general correspond to a 90° rotation of the original hyperbola; the two hyperbolas are generally different in shape.

A few other lengths are used to describe hyperbolas. Consider a line perpendicular to the transverse axis (i.e., parallel to the conjugate axis) that passes through one of the hyperbola's foci. The line segment connecting the two intersection points of this line with the hyperbola is known as the latus rectum and has length 

. The semi-latus rectum l is half of this length, i.e.

The focal parameter p is the distance from a focus to its corresponding directrix, and equals