To find the equation of an ellipse, let be any point of the ellipse and be the corresponding point on the directrix, as shown in the given diagram. Then by the definition of an ellipse, we have
It is clear from the given diagram that in the triangle , we have the relation given as
Using this relation in equation (i), we have
This is the equation of the ellipse whose center is at origin and foci lie on the X-axis. The lengths of the semi-major and semi-minor axes of this ellipse are and respectively.
If the foci lie on the Y-axis, then the graph is as shown in the given diagram. In this case the equation of the ellipse will be
NOTE: For this we use the relation