Example: Find the equation of the ellipse having center at origin, focus at and one vertex at the point .
Since the focus of the ellipse is at point , we take it as . Since the vertex of the ellipse is at point , by comparing we have .
For the ellipse we have the relation
Since the focus lies on the X-axis, the required equation of the ellipse is
Example: Find the equation of the ellipse with foci and , and the length of the major axis is .
The center of the ellipse is the midpoint joining the foci and , so the center of the ellipse can be found by using the midpoint formula. We have
Since the foci lie on the Y-axis with center , let the required equation of the ellipse be
Since the foci have the coordinates , , we have
Using this for the given foci , , we have
It is also given that . Putting these values in equation (ii), we have
Putting the values of and in equation (i), we have
This is the required equation of the ellipse.