Eccentricity e = c/a = sqrt(1 + b^2/a^2) uniquely characterizes the "openness" of a hyperbola. As e -> 1+, the hyperbola degenerates into a pair of parallel lines. As e -> infinity, the hyperbola opens wider, and asymptotes approach perpendicularity.

For the rectangular hyperbola (a = b), e = sqrt(2). This is the only conic with perpendicular asymptotes.

Focal chord properties: A chord through a focus is called a focal chord. If the endpoints of a focal chord have parameters theta_1 and theta_2, then tan(theta_1/2) * tan(theta_2/2) = (e-1)/(e+1) or -(e-1)/(e+1) depending on the focus.

The latus rectum (focal chord perpendicular to the transverse axis) has length 2b^2/a = 2a(e^2 - 1). Semi-latus rectum l = b^2/a. The focal distance of a point P(x_1, y_1) on the hyperbola is |ex_1 - a| and |ex_1 + a| from the nearer and farther foci respectively (for the right branch, x_1 > 0).

The definition |PF_1 - PF_2| = 2a is the absolute difference of focal distances. On the right branch: PF_1 - PF_2 = -2a (F_1 is the left focus), PF_2 - PF_1 = 2a if measured from the right focus.

Director circle: The locus of the point of intersection of perpendicular tangents to x^2/a^2 - y^2/b^2 = 1 is x^2 + y^2 = a^2 - b^2. This circle exists only when a > b; when a = b (rectangular hyperbola), perpendicular tangents meet at the center; when a < b, no real director circle exists.