In the derivation, the difference between the distances from any point P on a branch of the hyperbola to foci of the hyperbola is set equal to the constant value of .

A hyperbola is defined by the property that the absolute difference between the distances from any point on it to the two foci is constant. In the standard form, that constant is 2a, where a is the distance from the center to a vertex. Therefore, the correct choice is 2a.

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