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' = c/b. Key relationship: 1/e^2 + 1/e'^2 = 1, a result frequently tested in JEE.