Starting with Lens formula directly
$$ \frac1u + \frac1u = \frac1f $$
or in its Gauss form:
$$ (u-f)(v-f) = f^2, $$
how to recast this into the conics form using definition of eccentricity
$$ \frac{PF}{PD} = e\,, $$
at least as an approximation, using geometric optics where $PF,PD$ are focal and directrix distances of the conics curve ?
Edit1:
My query in other words is for finding relations between $(u,FD),(v,FP), (e, f, \mu) $ including any constants and linear approximations.
$$ \frac{ \sin i}{\sin r } = \mu ,\, \mu >1, =1$$ respectively for Snell Law refraction/reflection.
The formulas unify reflection and refraction swapping sign of $f$ or changing $\mu$ with above and corroborated by Fermat principle minimum optical path lengths.
Question is motivated on assumption of a valid unified derivation for conic shaped reflectors and refractors. It is a geometric optics question.
In Gaussian Optics (paraxial optics), the lens formula relates object distance and image distance with the focal length of the refracting (reflecting) surface. This focal length is related with the curvature of the surface and it has only meaning in that approximation. This means that even a conic has a focal length in the paraxial approximation. However, it only depends on the curvature of the surface despite this has eccentricity. I mean, the eccentricity has no role in the determination of the surface focal length.
If you think in the focal length as the distance for which a collimated light beam focus perfectly after refracting (reflecting) in the surface, then you should work with the optical path length definition along the bundle of rays belonging to the beam of light. This treatment expand the Gaussian Optics approximation and it is risky to talk about parameters defined and meaningful only within it.
Maybe I am missing something in your question.