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

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 _______.
2a is used in the equation because the _______ is a point on the hyperbola and it is the difference in the distances from this point to the foci.
This can be shown algebraically if we let a be the distance from the ______ to the vertex. Then, the difference of the distances is (c + a) – _______.