<s>[INST] 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
 [/INST] 
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 $\, \dfrac{\alpha^2}{a^2}+\dfrac{\beta^2}{b^2}+\dfrac{\gamma^2}{c^2}>(1+\lambda)^2$
</s>