# Equation of the Ellipse

To find the equation of an ellipse, let $P\left( {x,y} \right)$ be any point of the ellipse and $M\left( {\frac{a}{e},y} \right)$ the corresponding point on the directrix as shown in the given diagram, then by definition of ellipse, we have

It is clear from the given diagram that in the triangle, $FOB$, we have the relation given as

Using this relation in equation (i), we have

This is the equation of the ellipse whose centre is at origin and foci lie on the X-axis. The lengths of semi-major and semi-minor axes of this ellipse are $a$ and $b$ respectively.
If foci lie on the Y-axis, then its graph as shown in the given diagram. In this case the equation of ellipse will be

NOTE: For we use the relation ${a^2} - {b^2} = {a^2}{e^2}$