Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle C .

This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle B , as well as the envelope of the polar lines of the points on B . Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle

C ; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.

Quadratic equation

A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates (x ,

y ) in the plane,

provided that the constants A xx , A xy , A yy , B x,

B y , and C satisfy the determinant condition

This determinant is conventionally called the

discriminant of the conic section. [3]

A special case of a hyperbola—the degenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero:

This determinant Δ is sometimes called the discriminant of the conic section. [4]

Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in

Conic section#Eccentricity in terms of parameters of the quadratic form.

The center (x c , y c ) of the hyperbola may be determined from the formulaeformulae

In terms of new coordinates, ξ = x − x c and η =

y − yc , the defining equation of the hyperbola can be written

The principal axes of the hyperbola make an angle φ with the positive x-axis that is given by

Rotating the coordinate axes so that the x-axis is aligned with the transverse axis brings the equation into its canonical form