The hyperbola is a conic section with eccentricity e > 1, defined as the locus of a point whose distance from a fixed point (focus) bears a constant ratio e > 1 to its distance from a fixed line (directrix). The standard form is $x^2$/$a^2$ - $y^2$/$b^2$ = 1 with center at origin, transverse axis along the x-axis, and conjugate axis along the y-axis.

Key parameters: semi-transverse axis a, semi-conjugate axis b, relationship $c^2$ = $a^2$ + $b^2$ (note: plus, unlike the ellipse), eccentricity e = $\frac{c}{a}$ > 1. Vertices at (+/-a, 0), foci at (+/-c, 0), directrices at x = +/-a/e = +/-$a^2$/c. Length of latus rectum = 2$b^2$/a.

The rectangular hyperbola xy = $c^2$ (or $x^2$ - $y^2$ = $a^2$) has e = sqrt(2) and perpendicular asymptotes. Parametric form: (ct, c/t) for xy = $c^2$, and (asec(theta), btan(theta)) for the standard form.

Asymptotes y = +/-$\frac{b}{a}$x are the limiting positions of tangent lines as the point of tangency moves to infinity. The combined equation of asymptotes is $x^2$/$a^2$ - $y^2$/$b^2$ = 0. The conjugate hyperbola $x^2$/$a^2$ - $y^2$/$b^2$ = -1 shares the same asymptotes.

Tangent at ($x_1$, $y_1$): x*$x_1$/$a^2$ - y*$y_1$/$b^2$ = 1. Normal at ($x_1$, $y_1$): $a^2$*x/$x_1$ + $b^2$y/$y_1$ = $a^2$ + $b^2$. The condition for y = mx + c to be tangent: $c^2$ = $a^2$$m^2$ - $b^2$.

The reflection property: a ray directed toward one focus reflects off the hyperbola toward the other focus. This has applications in optics and satellite dish design. The difference of focal distances |$PF_1$ - $PF_2$| = 2a for any point P on the hyperbola.