https://en.wikipedia.org/wiki/Conic_section

Or in http://whistleralley.com/conics/conica/

Thanks, folks but these were all simply rhetoric - appeals to diagrams and "much talking".  I was looking for an algebraic derivation; none of these cut it. 

@Herb, which kind of answer would you wish? A reference (where can I find?) or an algebraic derivation of the equation? Or 2 in 1?

I am looking for an algebraic derivation, which obviously came after Apollonius. 

A one-page email should be sufficient. 

http://nebula.deanza.edu/~bloom/math43/ellipse-derivation.pdf

An ellipse has not an analytic equation in 3D.

@Herb, one can solve a system

(1)        x2 + y2 = kz2 , k>0

(2)        z = ax + by + d, d ≠ 0

The conic surface (cone of revolution) given in (1) has z-axis for its axis of symmetry, thus the plane cannot be parallel to it ... so one can take (2) as an equation of the plane.

We substitute "z" from (2) into (1). We obtain an equation (3) which is that of the ellipse [NOTE ADDED: naturally, for convenient a,b]. Unfortunately, (3) is not in the canonic form x2/A2 + y2/B2 =1, even not after completing the squares. It is due to the term 2kabxy. However, applying a rotation (which transforms our plane onto z=0 ) one can obtain an expected equation (3').

In 3D, the ellipse is defined by a system (1),(2) or (3),(2) or (3'),(z=0) ...

If you are looking for a more general form where the center of the ellipse is C(xc,yc,0) , use a translation moving the point (0,0,0) to C.

Don't you see no difference between your and my answer, Peter?

All above answers are answers to another question. For example, @Peter Breuer wrote "I took a cone ... and choose a simple plane ...". This way only can be proved that the intersection of picked up cone and plane is ellipse. But the question as it is stated asked another thing, namely, to prove that ellipse is intersection of a cone and a plane, that is, that every ellipse can be obtained as intersection of some cone and some plane. It is clear that before answering this question one must clarify what is meant under ellipse, i. e. to say from which definition of ellipse he want to derive the desired result by Apollonius.

Thanks, Guys.  The magic ingredient was the Equation of the Conical Surface. 

I already knew the 2-focii derivation but that does NOT talk about cones. 

Let me include some remarks about the former answers: All answers are giving a completely sufficient tips for an experienced user of "analytical geometry" about how one can derive equation of arbitrary ellipse in a form of  the intersection of a plane and the conic surface.  Indeed, the most close answer for the current question is to show how starting from a given equation of an ellipse (say, on the x)y plane:-) one might choose the equation for a cone such, the the ellipse is a line on that cone (as Aslanbek has suggested). This program s not performed yet. Bu,t perhaps Herb is not seeking so detailed algorithm. @Herb:  Are you? 

Regards, Joachim

@ Herb

Oops! our answers didn't appear chronically! Thus I see, that you really need more detailed tips. Let me start, however with a question: what do you mean under derivation of the foci, what data were  given for the derivation?

Regards, Joachim

My clue, Peter:

In the question "...an ellipse..." is (for me) "an ellipse in general". Your answer(s) were for "...the ellipse ..." You had a concrete case, one case from many.

Dear Peter,

It s almost an insult suggesting that such old teacher like me needs  the derivation of the equation of ellipse with given foci. My question was explicitely  directed to Herb in order to prepare an answer and/or suggestions regarding his knowledge about the derivation. 

By the way, you have written with regard Viera's answer:

"Err, isn't that what I did, a little up the page? In what sense is this new news?"

This suggests that you accept only, when other participants  write substantially new answer. Note the following: EVEN, if someone answer was less valuable, it could be (one should assume this apriori) of  a  form better understandable by the potential readers. I have recommended Viera's answer for this reason (this expresses my usefulness of her answer). Due to this, should I ask you in the spirit of  the following way: "Hello, Peter, do you see now how the aswers should be formulated? Your way of answering is only by examples and we need a general derivation. Didn't you know this?".

In order to avoid such unpolite remarks RG is since a year free of the possibility even to downvote the answers. Please, take into account that as long as answers are honestly oriented toward explanation the subject of the discussion, they are OK, even if they are weak, unclear in content or wrong. Obviously, in the last case everybody has right to mention the lack of correctness, obviously also in simple and polite claims like " this and this is incorrect since ..."

Best regards, Joachim

Yes, it would be helpful to see a 12-step proof of final elliptic equation (in 2D). 

Hopefully someone will find a  cone in 0xyz coordinates which crosses the 0xy-plane along the GIVEN ellipse  - be it  x2/a2 + y2/b2 =1  with GIVEN   a>0, b >0. The case of   a=b  is trivial; every such cone is determined by one of the following equations

  z= c - \sqrt{x2c2/a2  + y2c2/a2}, where  c>0  

OR  z = c + \sqrt{ x2c2/a2 + y2c2/a2} where  c

Don´t cry, Peter, don´t cry :)

I know that you know, that your solution in 1 concrete case would lead in a finite time to a complete solution. However, it is not automatically the case of each reader here. There are not only mathematicians.

So I entered a solution that could be useful for Herb. Simply, I wrote my answer as I felt it +/- correct and enough general. 

Dear Viera:  Where is your solution?  I cannot read invisible ink. 

Dear Herb,

it is on the first page, denoted (3) or (3´). Yes, written with an invisible ink :)

because I am sure you are able to substitute "z" from (2) into (1), and to simplify. 

Then I waited - will the author of the question be satisfied? or?

Dear all,

I like all answers and feel enthusiasm to share you the discussion. I consider the more general question:

Determine the surfaces S and G in R^3 such that their intersection is an ellipse E. In fact, there is an infinite number of surfaces S and G intersect in E. If we are interested in the case where one of the two surfaces, say G, is a plane, then G is uniquely determined by E. We need to pick three distinct points on E to find the plane G. Also there are infinite number of surfaces satisfy the required intersection E. As an example, for any point A not on G, join A to E via st.lines to determine an elliptic cone S. 

For a general analytic solution, assume that the equation of the ellipse E is given by       ( h(x,y,z)=0 ). Select three distinct points on E to find the equation of the plane G says: ax+by +cz+d=0, then the general surface S is determined by the equation:

S: (ax+by +cz+d)g(x,y,z)+mh(x,y,z)=0, where g is an arbitrary function defined over the points of the plane G and m is non-zero ( constant or function). One can choose g and m to determine the preferable form of the surface.

Best regards

"I do not imagine the angle of the cone sides make any difference at all, being determined by the distance to the ellipse." And I imagine.  This part of geometry is called stereometry. Though the problem when solved this way is difficult, but the idea is not cute. This is exacly what was suggested by Aslanbek, who writes: "to prove that ellipse is intersection of a cone and a plane, that is, that every ellipse can be obtained as intersection of some cone and some plane."

Peter:  We have all seen the Sliced-Cone diagram.

Why is it so difficult to see the algebraic equivalence? 

Good one, Peter:  Now add the final "6 ines" to get the canonical form

See https://en.wikipedia.org/wiki/Conic_section

Dear Mohammed: Thanks for your link (I had read it but it may be helpful to others).  We have all seen the pictorial impression but it is surprising how difficult it is (for moderns, at least) to prove the claim using analytic geometry and create an algebraic proof. 

I like this one https://en.wikipedia.org/wiki/Dandelin_spheres , although the proof does not use methods of analytic geometry.

Dear Viera:  Interesting info but surprising how this Old Result still challenges so many today. 

@Herb, the question was answered at the first page of this thread. From your comments I understand that it is not clear to you.

Enclosed is a clipping of http://www.mpcezanne.fr/Main/Maths/Colles2012/Polycopies/Geom3ConiquesQuadr.pdf

I hope the French language is not a problem. (valeur propre = eigenvalue,  vecteur propre = eigenvector)

A complete derivation of the equation is e.g. in

"promenades MATHEMATIQUES: histoire, fondements, applications"  (Frédéric Laroche, Ellipses, 2003) pp. 231-232, a beautiful book for a wide audience.