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Take a look at the following two definitions of ellipse:

For some fixed points $F_1,F_2$ and real number $2a>|F_1F_2|$ an ellipse is the locus of points $P$ such that $|F_1P|+|F_2P|=2a$.

For some fixed point $F$, line $d$ and number $e<1$ an ellipse is the locus of points $P$ such that $|FP|$ is $e$ times the distance from $F$ to $d$.

These two definitions can be easily shown equivalent using Dandelin spheres (which, in fact, also estabilishes that ellipse can be defined as a kind of conic section). However, for some time, I have been wondering if there is a way to show these definitions equivalent while "staying on the plane", i.e. without Dandelin spheres, cones etc.

My question here is: Is there any direct proof of equivalence of the above two definitions of ellipse? Also with "direct" I mean one without using the equation of an ellipse.

Thanks in advance.

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  • $\begingroup$ I didn't use the same variables you did, but I've proved this statement and the analogous one for hyperbolas in this question and answer. $\endgroup$ – Noble Mushtak Jun 21 '16 at 0:54
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$\frac{r1}{d1} =\frac{r2}{d2} = \frac{r1+r2}{d1+d2} = e$; Obviously $d_1+d_2$ is constant, so r1+r2 is also constant

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  • $\begingroup$ It's far from clear (to me) why existence of one directix implies existence of the second one (which is clearly what you need). It also doesn't show the other direction - if we know that $r1+r2$ is constant, we don't know if directix exists. $\endgroup$ – Wojowu Apr 26 '15 at 11:32
  • $\begingroup$ That's right, sorry I didn't say what I just posted just shows one direction, and I haven't come up with the other one since only knowing the sum $r1+r2$ it is hard to know where the vertical lines are( But in the above direction we know both the existence of vertical lines and focus, so the other direction is conceivably harder) $\endgroup$ – Xinyu Hua Apr 26 '15 at 12:16
  • $\begingroup$ As for the existence of the other directix, how about the Symmetric Property? $\endgroup$ – Xinyu Hua Apr 26 '15 at 12:30
  • $\begingroup$ If by "Symmetric Property" you mean the fact that ellipse has axis of symmetry along minor axis, then I'll tell you this: it's not obvious that curve defined using focus-directix definition has this property. $\endgroup$ – Wojowu Apr 26 '15 at 16:03
  • $\begingroup$ Also, how do you find the point which is supposed to serve as the second focus? If you think your proof can be saved, carefully reconsider what we know and what we don't know a priori. $\endgroup$ – Wojowu Apr 26 '15 at 16:05

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