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The focus-directrix definition of a conic says that a (non-circular) conic is a collection of points such that there exists a line $\ell$ called a directrix and a point $F$ called a focus outside the line such that conic is the locus of all points $P$ such that the ratio of the distance from $P$ to $F$ divided by the distance from $P$ to $\ell$ is constant. This constant is called the eccentricity of the conic and the distance from the focus to the directrix is called the focal parameter. I have two related questions about well-definedness:

  1. Considering some conics have more than one focus-directrix pair, how do we know that the eccentricity and focal parameter are well-defined? That is, why do they have the same value for all focus-directrix pairs?

  2. Now, one might point to the fact that we know that a parabola has one focus-directrix pair, and that ellipses and hyperbolas have two, but I am not sure how to prove that. I know how to prove their existence, but why does a parabola have at most one focus-directrix pair, and why is there at most two for ellipses and hyperbolas?

The first question is important to me because we refer to "the" eccentricity and "the" focal parameter while I have been unable to find a source that proves that they are well-defined. As for the second question, there are several concepts such as the center of an ellipse, linear eccentricity, the latus rectum, and more that are easier to define or make well-defined if it is clear exactly how many focus-directrix pairs exist and where they lie.

An answer or a reference to a source would be greatly appreciated.

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  • $\begingroup$ Look up Dandelin spheres. $\endgroup$ – amd Apr 1 '20 at 19:51
  • $\begingroup$ I am aware of the concept of Dandelin spheres, but I am not aware of any proof involving them that answers my two questions. $\endgroup$ – Favst Apr 1 '20 at 22:02
  • $\begingroup$ Are you aware of how to construct the directices of a conic using its Dandelin spheres? $\endgroup$ – amd Apr 1 '20 at 22:06
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Answer to 1.

Consider an ellipse with focus $F$ and directrix $d$. The line through $F$ perpendicular to $d$ is an axis of symmetry for the ellipse (this is obvious from the definition): let $H$ be the point where this line meets the directrix.

Taken any point $P$ on the ellipse, if $y=PK$ is its distance from $d$, we know that its distance from the focus is $PF=ey$, where $e<1$ is the eccentricity. Setting $x=KH$ and $p=FH$ we then obtain from Pythagoras' theorem the relation: $$ (y-p)^2+x^2=(ey)^2, \quad\text{that is:}\quad x^2=2py-(1-e^2)y^2-p^2. $$ For a given value of $x$ two solutions $y$ and $y'$ are possible, corresponding to two different points $P$ and $P'$ on the ellipse, with $$y+y'={2p\over1-e^2}.$$

This implies that the ellipse has a second axis of symmetry $s$, parallel to the directrix and at distance $p/(1-e^2)$ from it. The intersection of the axes of symmetry is the center $O$ of the ellipse. It follows that $F'$ and $d'$, reflections about $s$ of $F$ and $d$, also work as focus and directrix, with the same eccentricity $e$.

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From the above equations one can also find the lengths of the diameters lying on the axes: $$ AA'={2ep\over1-e^2},\quad BB'={2ep\over\sqrt{1-e^2}}, \quad\text{whence:}\quad {BB'\over AA'}=\sqrt{1-e^2}<1. $$ In particular, $BB'<AA'$, a result we'll need later.

The case of a hyperbola is similar and is left to the reader.

Answer to 2.

As noticed above, the line $a$ passing through the focus and perpendicular to the directrix is an axis of symmetry for the conic section. This means that all chords perpendicular to $a$ are also bisected by $a$.

Suppose now that a conic section has another pair focus/directrix, in addition to the usual ones. This would imply another axis of symmetry $a'$, bisecting all chords perpendicular to it. But this is not possible for the reasons explained below.

1) In a parabola, the midpoints of parallel chords are aligned along a line parallel to $a$ but not perpendicular to the chords (unless of course this line is $a$ itself).

2) In an ellipse or hyperbola, the midpoints of parallel chords are aligned along a line passing through the center of the conic. This line is perpendicular to the chords only if it is the same as $a$ or the secondary axis $s$ discussed above.

To exclude the possibility $s=a'$, just observe that line $s$ doesn't meet the conic in the case of a hyperbola, while it meets an ellipse at the endpoints $BB'$ of the minor axis, which is always shorter than major axis $AA'$ (unless the ellipse is a circle). Hence any ellipse having its focus on $s$, directrix perpendicular to $s$ and passing through $B$, $B'$, cannot pass through $A$, $A'$. This concludes the proof.

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