I have a cone which is passing through a plane. The cone is not perpendicular to the plane, so the intersection area between the cone and the plane will not be circular but an ellipse. The cone will appear to be perpendicular to the plane in the yz-plane, so the positive and negative z-radii will be equal.
Given the xy-plane cross section of the cone that looks as follows:
The negative x-radius, $\overline{SI}$, is shorter than the positive x-radius, $\overline{IL}$. My question is: Is the z-radii length equal to the average length of $\overline{SI}$ and $\overline{IL}$, that is the length of $\overline{SL}$ over 2?
Short answer: no.
Medium length answer: if the length of the $z$ radius was that of half of $SL$, this would mean that the intersection is a circle, which you know it is not...