Any conic section symmetric about the origin satisfies the equation
![\[ \lVert X - F \rVert = | eX \cdot N - a| \qquad \text{where} \qquad a= ed + eF \cdot N. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a8193404b2b4f9cb355561027dcd037c_l3.png "Rendered by QuickLaTeX.com")
Using this, prove that if the conic section is an ellipse then we have
![\[ \lVert X - F \rVert + \lVert X + F \rVert = 2a. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-8385e9b23483fe3483f66fdd1b663012_l3.png "Rendered by QuickLaTeX.com")
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.
let X=(x,y) and N=(1,0). Then XN=x and |ex|<=a. From that we have |eXN-a|=|ex-a|=a-ex. If we plug -F there to get the distance from the other focal point, and sum the equalities, we get 2a.