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?
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).