This is an exploration of what is involved when you try to find the general function for the tangent to an ellipse. Since an ellipse is not a function, a piecewise function is required in order to determine the general function of the tangent.
It is possible to determine each of the necessary functions individually, but with a little careful thought and consideration of concepts such as symmetry, it is possible to avoid doing this, resulting in a more complex understanding of what is going on.

What happens to the tangent when x = 2 or x=-2?
This applet only works for the given ellipse. Could you create a similar applet which works for any ellipse at all, i.e., could you find the general function for the tangent to the general equation for an ellipse?