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Prove that the sum of distances from a point on a ellipse to its foci is constant

by RoRi

Any conic section symmetric about the origin satisfies the equation

    \[ \lVert X - F \rVert = | eX \cdot N - a| \qquad \text{where} \qquad a= ed + eF \cdot N. \]

Using this, prove that if the conic section is an ellipse then we have

    \[ \lVert X - F \rVert + \lVert X + F \rVert = 2a. \]

This can be interpreted to say that the sum of the distances from a point on an ellipse to the foci is a constant.

Proof. Incomplete.

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