E, F are the ellipse foci, EF meets $ \epsilon$ at A, B, AB is the ellipse major axis. The cone with directrix $ \epsilon$ is right $ \Longrightarrow \exists$ two Dandelin spheres $ \mathcal S_e,\ \mathcal S_f$ tangent to the cone, centered at $ I_e, I_f$ on its rotation axis ($ P,\ I_e,\ I_f$ are collinear), and touching the plane $ (\epsilon)$ at E, F. $ EI_e,\ FI_f \perp (\epsilon),$ $ PI_eI_f$ is in a plane $ \nu \perp (\epsilon),$ such that $ AB \equiv EF \equiv \nu \cap (\epsilon).$ The plane $ \nu$ cuts the spheres $ \mathcal S_e,\ \mathcal S_f$ in the incircle and P-excircle $ (I_e),\ (I_f)$ of the $ \triangle PAB.$ The incircle $ (I_e)$ touches PA, PB at U, V. $ PB - PA = (PV + VB) - (PU + UA) = EB - EA =$ $ = EF + FB - EA = EF = \text{const}$ $ \Longrightarrow P$ is on a hyperbola $ h \in \nu$ with foci A, B and major axis EF.