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
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.:
vertex
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) – .:
center
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) – .: