# Equivalence of focus-focus and focus-directix definitions of ellipse without leaving the plane

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.

$\frac{r1}{d1} =\frac{r2}{d2} = \frac{r1+r2}{d1+d2} = e$; Obviously $d_1+d_2$ is constant, so r1+r2 is also constant
• 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. – Wojowu Apr 26 '15 at 11:32
• 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) – Xinyu Hua Apr 26 '15 at 12:16