This ellipse must be tangent to at least 4 lines and it must intersect the other lines. I've tried to use ellipses that are parallel to the x- and y-axis. I've done this by transforming the equation of this lines into a dual space:
l: px + qy + r = 0 with p2 + q2 = 1
For a fixed ellipse, this can be transformed into:
p2a2 + q2b2 - (px0 + qy0 + r)2 = 0
This has worked, but I'm just wondering how to find an ellipse that isn't parallel to the x- and y-axis in an elegant way. All types of suggestions are welcome.
Let's say that you have four lines. They determine a quadrilateral. Take the smallest of its two diagonals. The smallest ellipse touching all four sides is the degenerate ellipse coinciding with said diagonal, i.e., whose minor axis is zero, and whose major axis is the aforementioned diagonal. To visualize this, use GeoGebra.