The two standard orientations of a hyperbola are: Horizontal: $x^2$/$a^2$ - $y^2$/$b^2$ = 1 (transverse axis along x) and Vertical: $y^2$/$a^2$ - $x^2$/$b^2$ = 1 (transverse axis along y). In both cases, $c^2$ = $a^2$ + $b^2$.

Parametric representation for $x^2$/$a^2$ - $y^2$/$b^2$ = 1: x = asec(theta), y = btan(theta), using the identity $sec^2$(theta) - $tan^2$(theta) = 1. Alternatively, x = acosh(t), y = bsinh(t) using $cosh^2$(t) - $sinh^2$(t) = 1.

For the rectangular hyperbola xy = $c^2$: parametric form is (ct, c/t) where t is the parameter. The tangent at parameter t is x/t + yt = 2c. The normal is $xt^3$ - yt = c($t^4$ - 1).

Shifted hyperbola: (x-h)^2/$a^2$ - (y-k)^2/$b^2$ = 1 has center (h, k), vertices at (h +/- a, k), foci at (h +/- c, k). All formulas translate by (h, k).

The auxiliary circle $x^2$ + $y^2$ = $a^2$ relates to the hyperbola through the eccentric angle: if P(asec(theta), btan(theta)) is on the hyperbola and Q(acos(theta), asin(theta)) is on the circle, theta connects them. This is less commonly used than the ellipse auxiliary circle, but appears in theoretical problems.

Conjugate hyperbola: -$x^2$/$a^2$ + $y^2$/$b^2$ = 1 has semi-transverse axis b (along y), semi-conjugate axis a (along x), eccentricity e' = $\frac{c}{b}$. Key relationship: 1/$e^2$ + 1/e'^2 = 1, a result frequently tested in JEE.