To prove the equation of a hyperbola, let be any point of the hyperbola and be the corresponding point on the directrix as shown in the given diagram. Then by the definition of a hyperbola, we have
From the diagram of the hyperbola it is clear that
Putting this value in equation (i), we have
This is the equation of the hyperbola whose center is at origin and the foci lie on the X-axis. The lengths of the transverse axis and conjugate axis of this hyperbola are and respectively.
If the foci lie on the Y-axis, then its graph is as shown in the given diagram. In this case the equation of the hyperbola will be