Eccentricity is invariant for ellipse defined by intersection between plane and ellipsoid [can't be correct]?? When a plane intersects a sphere, the intersection is always a circle, due to rotational symmetry etc.
However, if a plane intersects an ellipsoid( say, the rotation of $$(\frac{x}{a})^2+(\frac{y}{b})^2=1$$ about the x axis) it forms ellipses of different eccentricities. It makes sense that every plane parallel to the x-z plane intersects to form a circle.
Similarly, (i suspect that) all planes  parallel to the xy plane form an ellipse of eccentricity of $b/a$. 
Are all eccentrities of a planar intersection with this ellipsoid bounded by these two values?
I took a look at a generic ellipsoid:
$$(\frac{x}{a})^2+(\frac{y}{b})^2+(\frac{z}{c})^2=1$$
and a general plane: $$fx+gy+hz=1$$.
At first, I tried something weird, but I just wanted to get a 2-variable expression for the answer, so I used substitution to get the solution set:
$$(\frac{x}{a})^2+(\frac{y}{b})^2+(\frac{1-fx-gy}{ch})^2=1$$
But then isn't the definition of eccentricity: c/a, or:
$$eccentricity^2=1-(\frac{b}{a})^2$$.
But then the eccentricity is invariant, when looking at the generic plane? This makes no sense to me, does anyone have any insights?
 Sorry, I have not done this type of math in a very long time.
 A: Simply substituting the formula of the plane into that of the ellipse is not going to work as expected. Computing the eccentricity from a coordinate representation of an ellipse requires the underlying basis to be orthonormal. And while for most planes, each point can be described by $x$ and $y$ coordinate, with the $z$ coordinate implied by the equation of the plane, the projections of the $x$ and $y$ vector onto that plane are in general not orthonormal.
So I'd start by picking two orthogonal unit length vectors in the plane, and then expressing the ellipse with respect to these. Then you will see that the principal axes depend very much on the choice of plane.
A: This isn't an answer. What I was thinking of as a counterexample is the parametric plot of:
$
   \left\{\frac{4}{3} \sqrt{2} \cos (t),\frac{2}{3} \sqrt{2} \sin (t+45
    {}^{\circ})\right\}
$
which looks like this:

so the x and y intercepts are not the lengths of the major and minor axes.
Unfortunately, I don't think this figure is actually an ellipse, so my comment was incorrect.
