Standard proof that ellipses satisfy constant sum property

This proof of the constant sum property assumes that an ellipse is a figure with the standard rectangular coordinate equation.

We are given an ellipse in the form of the equation

We may assume that a > b. (If not, reverse the roles of x and y by switching the names a and b.) Define The points that will serve as foci are (–c, 0) and (c, 0). Choose any point (x, y) satisfying so that Then the squares of the distances from (x , y ) to the foci are

It follows that the distances from (x , y) to the foci are

Hence their sum is 2a .