Compute Curve Given Implicitly

So I have a curve given implicitly by the two equasions:

So I've defined functions such that:

For the NEW curve (which I'll call C for argument's sake, and I appreciate here a sketch might be useful, but I don't know how to do one online), I'll let be the "centre" of the curve (which a sketch will show you is an ellipse/circle) and will define the vectors to be an orthonormal basis in this new curve.

I'm doing this to make it easier to paramatrise, as we will effectively take it down to 2-Dimensions, in this new basis.

Now for the bit I'm stuck on. I'd LIKE to substitute into or , but it just doesn't work. I don't get the equasion of a circle (our lecturer told us it would be a circle, because we have to arc length parametrise it later.

I've tried "swapping round" the values in , but I'm sure they're right, and that gives me something even worse than what I get the other way. Any help?