Asymptotes of $x^2$/$a^2$ - $y^2$/$b^2$ = 1 are y = $\frac{b}{a}$x and y = -$\frac{b}{a}$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$\frac{b}{a}$.

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 $x^2$ - $y^2$ = 2$c^2$ 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 $a^2$b^$\frac{2}{a^2+b^2}$ = $a^2$$b^2$/$c^2$.

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