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Prove that the points on each branch of a hyperbola satisfy a property

by RoRi

Prove that on each branch of a hyperbola the difference

    \[ \lVert X - F \rVert - \lVert X + F \rVert \]

is a constant.

Proof. Incomplete.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):