Are focus and directrix for a conic section curve unique?

Question

Question. Can a conic section curve have two distinct pairs of focus and directrix?

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Attempt. I cannot think of a rigorous and logical way to convince myself of the uniqueness. But for me it is like two degrees of freedom (focus and directrix) define a single degree of freedom (a curve based on the constant eccentricity); intuitively for me, it might not be unique?

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Comment. I do not need the answer to be rigorous; just some framework or hint would satisfy. Thank you!

Answer 1

We can always find the unique pair of Dandelin spheres from the conic section without ambiguity.

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Each Dandelin sphere touches both the right circular cone and the conic section on the corresponding focus.

See also detail discussion with animation in the link here and another post of mine about focal conics and confocal quadrics here.

For a general conic, namely

$$0= \begin{pmatrix} x & y & 1 \\ \end{pmatrix} \begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix} \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$$

Its foci $z=x+yi$ are given by the complex quadratic, namely

$$Cz^2-2(G+Fi)z+(A-B)+2Hi=0$$

where capital letters $A,B, \ldots$ represent co-factors of the corresponding entries of small letters $a,b, \ldots$ of the matrix respectively. That is

$$ \begin{pmatrix} A & H & G \\ H & B & F \\ G & F & C \end{pmatrix} = \begin{pmatrix} bc-f^2 & fg-ch & fh-bg \\ fg-ch & ac-g^2 & gh-af \\ fh-bg & gh-af & ab-h^2 \end{pmatrix}$$

See another post for finding the principal axes, etc.

Answer 2

Commenters have pointed out that ellipses and hyperbolae have two focus/directrix pairs. But I assume that's not what the OP is after.

As another answer points out, given a conic you can uniquely determine its foci, center, axes, directrices, etc.

But OP would like to reconcile this with "two degrees of freedom (focus and directrix) define a single degree of freedom (a curve based on the constant eccentricity)"

For a given conic and its focus, the directrix is the polar of the focus with respect to the conic. So the focus/directrix pair are really a single degree of freedom in this context.