parameterized form of rotated ellipse?

Use a 2D rotation matrix to come up with, for example (many variations possible):

{ Cos(a)Sin(t) - kSin(a)Cos(t), Sin(a)Sin(t) + kCos(a)Cos(t) }

where t is the parameter variable, k is the ratio of the ellipse axes, and a is the angle of rotation. You can easily convert the trigonometric expressions into algebraic form, since a and t are merely parameters.

Edit: An example of an algebraic explicit function of the form f(x,y) = 0 that rotates as a (not an angle but a generalized parameter) changes would be:

A'x^2 + B'y^2 + C'xy + D' = 0

where

A' = A^2 - 2A^2 a^2 + 4B^2 a^2 + A^2 a^4

B' = B^2 + 4A^2 a^2 - 2B^2 a^2 + B^2 a^4

C' = -4A^2 a + 4B^2 a + 4A^2 a^3 - 4B^2 a^3

D' = -A^2 B^2 - 2A^2 B^2 a^2 - A^2 B^2 a^4