Circle configuration at \(R=|h_{\max}|\) in the negative-energy parabolic case

Determine the relative position of the foci-circle and foci-caustic circle when \(-f<h_{\max}<0\) and \(R=|h_{\max}|\) for the parabolic boundary enclosing \(F'\).

Background

For the parabolic boundary enclosing $F'$, the paper classifies the relative positions of the foci-circle and foci-caustic circle across several ranges of the parameter $h_{\max}=E/(mg)$. In Case 5, where $-f<h_{\max}<0$, the authors state the behavior for $|h_{\max}|<R<h_{\max}+2f$ and for the endpoint $R=h_{\max}+2f$, but do not resolve the intermediate boundary value $R=|h_{\max}|$.

The unresolved value is specifically flagged in a margin note and concerns whether the two circles are disjoint, tangent, intersecting, or nested at that parameter.