To prove the equation of a hyperbola, let be any point of the hyperbola and the corresponding point on the directrix as shown in the given diagram, then by definition of hyperbola, we have

From the diagram of hyperbola it is clear that

Putting this value in equation (i), we have

This is the equation of the hyperbola whose centre is at origin and foci lie on the X-axis. The lengths of transverse axis and conjugate axes of this hyperbola are

and

respectively.

If foci lie on the Y-axis, then its graph as shown in the given diagram. In this case the equation of hyperbola will be

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