Asymptotes of x^2/a^2 - y^2/b^2 = 1 are y = (b/a)x and y = -(b/a)x, or combined: x^2/a^2 - y^2/b^2 = 0. The asymptotes pass through the center and have slopes +/- b/a. The angle between asymptotes is 2*arctan(b/a).

Key property: The equation of the hyperbola, the combined equation of its asymptotes, and the conjugate hyperbola differ only in the constant term. Hyperbola: S = 1, Asymptotes: S = 0, Conjugate: S = -1, where S = x^2/a^2 - y^2/b^2.

The conjugate hyperbola -x^2/a^2 + y^2/b^2 = 1 shares the same asymptotes and center but has its transverse axis perpendicular to the original. If the original has eccentricity e, the conjugate has eccentricity e' where 1/e^2 + 1/e'^2 = 1.

The rectangular hyperbola xy = c^2 (rotation of x^2 - y^2 = 2c^2 by 45 degrees) has asymptotes as the coordinate axes (x = 0 and y = 0). This form is particularly convenient for problems involving perpendicular asymptotes.

Properties involving asymptotes: (1) Any line parallel to an asymptote meets the hyperbola at exactly one finite point. (2) The perpendicular distance from a focus to an asymptote equals b. (3) The product of perpendicular distances from any point on the hyperbola to the two asymptotes is a^2b^2/(a^2+b^2) = a^2b^2/c^2.

JEE application: Finding the equation of a hyperbola given its asymptotes -- the hyperbola equation is the asymptote equation +/- constant.