Eccentricity e = $\frac{c}{a}$ = 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$\frac{theta_1}{2}$ * tan$\frac{theta_2}{2}$ = $\frac{e-1}{(e+1)}$ or -$\frac{e-1}{(e+1)}$ depending on the focus.

The latus rectum (focal chord perpendicular to the transverse axis) has length 2$b^2$/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.