TITLE: Orthogonality of major and minor axes of an ellipse
QUESTION [5 upvotes]: How would one go about proving that the major and minor axes of an ellipse are perpendicular bisectors of each other? I've read several proofs of expressions for the ellipse in cartesian and polar coordinates given the definition of the ellipse as the collection of points where the sum of the distance from two points is constant, but they always just assume that it is true. Visually it looks true, but is there an argument that follows simply from the definition?

REPLY [5 votes]: As per the principal axis theorem, the principal axes are given by the eigenvectors of the quadratic form that defines the ellipse. This quadratic form is symmetric, and therefore (by the spectral theorem) its eigenvectors are orthogonal. $\blacksquare$
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Naturally, we assume that the eccentricity is non-zero; otherwise, the claim is vacuous.