To find the equation of an ellipse, let be any point of the ellipse and 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, , 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 and 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