# Equation of the Hyperbola

To prove the equation of a hyperbola, let $P\left( {x,y} \right)$ be any point of the hyperbola and $M\left( {\frac{a}{e},y} \right)$ 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 $a$ and $b$ 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