Example: Find an equation of the ellipse having centre at origin, focus at and one vertex at the point .
Since the focus of an ellipse is at point , so we take it as . Since the vertex of an ellipse is at the point , so by comparing we have .
For ellipse we have the relation
Since the focus lies on the X-axis, so the required equation of ellipse is
Example: Find an equation of the ellipse with foci and , also length of major axis is .
Centre of the ellipse is the midpoint joining the foci and , so the centre of ellipse can be find using midpoint formula, we have
Since the foci lie on Y-axis with centre , so let the required equation of ellipse will be
Since the foci have the coordinates , , so 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 ellipse.