Proof 2 of rectangluar coordinate equation

We assume that an ellipse is a figure that satisfies the constant sum property.

Let the x-axis be the major axis, and the y-axis be the minor axis. Let b denote the length of the semi-minor axis, and define c to be the positive square root of a ² – b ². It follows that the foci are at (±c, 0).

The distance formula for rectangular coordinates allows us to state the fact that the sum of distances from the foci is the constant 2a as

Transposing one of the radicals and squaring, then repeating the process, yields

b² x² + a² y² = a² b²,

which is equivalent to the desired equation,