I Converse of focus-directrix property of conic sections

AI Thread Summary
The discussion explores the focus-directrix property of conic sections, emphasizing that the distance from a fixed point (focus) is proportional to the distance from a fixed line (directrix), with eccentricity as the proportionality constant. The inquiry centers on whether the converse is true: if a locus of points adheres to the focus-directrix property, does it necessarily represent a conic section derived from a cone? Participants are encouraged to compare the equations of general conic sections with those of plane curves satisfying the focus-directrix property. The conversation also seeks to clarify the conditions that define the equations for conic sections. Establishing this equivalence could deepen the understanding of conic sections in geometry.
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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?
 
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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.
 
pasmith said:
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?
 
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