There are several potential definitions of an ellipse and it is not trivial to prove that these definitions are equivalent.
1 Affine transformation of a circle.
x2/a2 + y2/b2 = 1
defines an ellipse with width 2a and height 2b centered at the origin.
2 Kepler’s equation
r = 1/(1 − e sinθ)
in polar coördinates defines an ellipse with a focus at the origin and eccentricity e.
3 String Construction
Another construction is the set of points where the sum of two distances to the two foci is constant.
4 Conic Section
Then there is the definition the intersection of a cone and a plane.

No two of these definitions are easily proven to be equivalent. The Dandelin spheres are a fairly easy connection between definitions 3 and 4.