Given two circles (defined by center and radius), how do I find the smallest ellipse that encloses both of them? I.e. I search the green ellipse in the picture below.
The ellipses can be considered axes aligned if this simplifies things. The end goal is to classify points on whether they are in the ellipse or not. So a final representation suitable to the form used in this question will be preferred.
Join the two centers and extend the line. wherever it cuts the two circles at their outer extremes are the points which are crucial say $A$ and $B$. take the mid point of these two points. for simplicity consider this point as the origin.
now consider an ellipse with the major axis as the line segment $AB$ and call this as $x$-axis. you need to find the minor axis and you are done. the equation of an ellipse is
$$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$$
here $a$ is known and $b$ is to be found. find the radius of the osculating circle of this ellipse at the points $A$ and $B$, which is of course the same numerical value (why?)
then equate this numerical value to the maximum of the two radii. this will give you an equation in $b$ which you can solve.