I have the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ and the plane $n_xx+n_yy+n_zz=0$. They intersect along an ellipse.
1) What is the formula of the ellipse, and
2) What is the position of each of the foci?
I've tried substituting $z=\frac{-n_x x-n_y y}{n_z}$ (and similarly, the values of $x$ and $y$ from the equation of the plane) into the equation of the ellipsoid, but this gives me the formula for an ellipse which lies in the x-y plane. This shouldn't be the formula of the ellipse since the actual formula should lie in the plane $n_xx+n_yy+n_zz=0$. My understanding is that by substituting this value of $z$, I am obtaining the projection of the actual ellipse onto the x-y plane.
To obtain the position of the foci of the actual ellipse, I'm thinking of finding the foci of the ellipses formed by the substitution of the values of $y$ and $z$ from the equation of the plane and finding the $x, z$ and $y$ coordinates of the foci respectively, combining them to give the actual foci of the ellipse. Would this be the correct approach?
Edit:As Kaster pointed out, the substitution alone doesn't give the equation of the ellipse - so most of the penultimate paragraph is incorrect.
For the first part, you are using coordinate the coordinates (x,y) as the coordinates on the plane, but as you note, these are projection so they are not normalized coordinates. To fix this, you can scale x by $ \sqrt{ 1 + \frac{n_x^2}{n_z^2}} $ and y similarly.