Compute Curve Given Implicitly

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

$\displaystyle \frac{x}{a}^2 + \frac{y}{b}^2 + \frac{z}{c}^2 = 1$

$\displaystyle a \sqrt{b^2 - c^2}z = c\sqrt{a^2 - b^2}$

So I've defined functions such that:

$\displaystyle F_{1}(x,y,z) = \frac{x}{a}^2 + \frac{y}{b}^2 + \frac{z}{c}^2 - 1$

$\displaystyle F_{2}(x,y,z) = a\sqrt{b^2 - c^2}z - c\sqrt{a^2 - b^2}$

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 $\displaystyle p_{0}$ be the "centre" of the curve (which a sketch will show you is an ellipse/circle) and will define the vectors $\displaystyle e_{1}, e_{2}$ 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.

$\displaystyle p_{0} = (0,0,0)$

$\displaystyle e_{1} = (a\sqrt{b^2 - c^2}, 0, c\sqrt{a^2 - b^2})$

$\displaystyle e_{2} = (0,1,0)$

Now for the bit I'm stuck on. I'd LIKE to substitute $\displaystyle p_{0} + \lambda e_{1} + \mu e_{2}$ into $\displaystyle F_{1}$ or $\displaystyle F_{2}$, 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 $\displaystyle e_{2}$, but I'm sure they're right, and that gives me something even worse than what I get the other way. Any help?