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 = c/a > 1. Vertices at (+/-a, 0), foci at (+/-c, 0), directrices at x = +/-a/e = +/-a^2/c. Length of latus rectum = 2b^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 = +/-(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): xx_1/a^2 - yy_1/b^2 = 1. Normal at (x_1, y_1): a^2x/x_1 + b^2y/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.