Converse of focus-directrix property of conic sections

[arham_jain_hsr]

TL;DR Summary

If the locus of some points follows the focus-directrix property, then is the curve ALWAYS the cross-section of a cone?

In my recent study of Conic Sections, I have come across several proofs (many of those comprise Dandelin spheres) showing that the cross-section of a cone indeed follows the focus-directrix property:

"For a section of a cone, the distance from a fixed point (the focus) is proportional to the distance from a fixed line (the directrix), the constant of proportionality being called the eccentricity."

But, in order to truly establish equivalence between the two definitions of the conic sections, I am curious to know whether the converse of this is also true. That is, if the locus of some points follows the focus-directrix property, then is the curve ALWAYS the cross-section of a cone?

 

[pasmith]

You can calculate the equation of a general conic section in the plane of the section.

You can calculate the equation of a general plane curve that satisfies the focus-directrix property.

Now compare the two.

 

[arham_jain_hsr]

You can calculate the equation of a general conic section in the plane of the section.

You can calculate the equation of a general plane curve that satisfies the focus-directrix property.

Now compare the two.

For the focus-directrix property, the equations are fairly obvious, that the point should have a certain ratio with the focus and directrix. What are the conditions that govern the formation of equations for the section of a cone definition?