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There are at least $4$ definitions for ellipse:

1) Scaling a circle: $(x/a)^2+(y/b)^2=1$

2) Sum of distances from two points is constant $PF_1+PF_2=\text{constant}$

3) Focus, Directrix: $e=\dfrac{\text{distance between point and Focus}}{\text{distance between point and Directrix}}$
4) Cutting a cone with a slanted plane.

I managed to see the equivalence between #2 and #4 because of this awesome video. From then I've been trying to see a geometric connection between #1 and #2, but no luck yet. Using coordinate geometry & algebra easy to establish $ \#2\Rightarrow \#1$. But how to see $\#1\Rightarrow \#2$? Any help?

AgentS
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    It is much easier to show the equivalence of #1 and #2. –  Dec 06 '19 at 15:12
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    @YvesDaoust just realized that and changed the question. I'm also trying it on paper. Thanks:) – AgentS Dec 06 '19 at 15:13
  • @YvesDaoust I could show it easily with algebra. Actually that's how all high school textbooks derive the equation for ellipse. However this is not that satisfying... Pretty sure there is also a nice geometric way to reason as scaling a circle is – AgentS Dec 06 '19 at 15:15
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    When considering the focus-based definition, you can say that the circle is a special case of an ellipse, with two coincident foci. But no continuous deformation of a circle can split the center in two foci. –  Dec 06 '19 at 15:27
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    In $(x/a)^2+(y/b)^2=1$, as $a\to b$, the distance between foci approaches $0$. So we can assume a circle has two foci lying at the center. Then when we stretch the circle(disc) these two focii also stretch... oh I see your point. The origin $(0,0)$ stays fixed under scaling XD – AgentS Dec 06 '19 at 15:31
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    Why can't you reverse your steps to get 1 to 2? Alternatively, given the equation of the ellipse, you know what the foci are, so prove that the distances sum to a constant. – Calvin Lin Dec 06 '19 at 15:39
  • @CalvinLin thanks, sure I can but scaling a circle is independent of coordinate system. I'm still trying to see how scaling gives ellipse geometrically.. – AgentS Dec 06 '19 at 16:33
  • Looks Dandelin spheres show that result also. In that linked video in the post, the ellipse in the plane approaches circle when the boundary of ellipse bisects the line from other two circles everywhere. Not sure if it makes sense but it is same as putting $a=b$ in $(x/a)^2+(y/b)^2=1$ – AgentS Dec 06 '19 at 16:40

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A circle is an ellipse, but an ellipse is not a circle. U can show that the distance between two foci is zero, that implies that scaling give a circle. U also could prove that the distance between any point at circumference and a foci is constant (circle equation).

Bswan
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