equation of the plane passing through the intersection of two ellipsoids I want to find the equation of the plane passing the intersection of two intersecting ellipsoids. the intersection of two ellipsoids is always a ellipse. I need to find the equation of the planar surface containing this ellipse.
Does have anyone any idea in this matter?
thanks in advance for your any response


Only two similar and translated ellipsoids guarantee planar ellipse section:

For $\lambda > 0$,
\begin{align*}
  \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}
  &= 1 \tag{1} \\
  \frac{(x-\alpha)^2}{a^2}+\frac{(y-\beta)^2}{b^2}+\frac{(z-\gamma)^2}{c^2}
  &= \lambda^2 \tag{2} \\
\end{align*}
$(1)-(2)$,
$$\frac{2\alpha x}{a^2}+\frac{2\beta y}{b^2}+\frac{2\gamma z}{c^2}=
1-\lambda^2+\frac{\alpha^2}{a^2}+\frac{\beta^2}{b^2}+\frac{\gamma^2}{c^2}$$
which is the radical plane of the two ellipsoids.
The centre of section is:
$$
\frac
{1-\lambda^2+\dfrac{\alpha^2}{a^2}+\dfrac{\beta^2}{b^2}+\dfrac{\gamma^2}{c^2}}{2\left(
    \dfrac{\alpha^2}{a^2}+\dfrac{\beta^2}{b^2}+\dfrac{\gamma^2}{c^2}
  \right)}
\begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix}
$$
Note briefly:
No intersections if $\, \dfrac{\alpha^2}{a^2}+\dfrac{\beta^2}{b^2}+\dfrac{\gamma^2}{c^2}>(1+\lambda)^2$
