# Example of Finding the Equation of an Ellipse

Example: Find the equation of the ellipse having center at origin, focus at $\left( {3,0} \right)$ and one vertex at the point $\left( {5,0} \right)$.

Since the focus of the ellipse is at point $\left( {3,0} \right)$, we take it as $ae = 3$. Since the vertex of the ellipse is at point $\left( {5,0} \right)$, by comparing we have $a = 5$.

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 $\left( {0, - 2} \right)$ and $\left( {0, - 6} \right)$, and the length of the major axis is $8$.

The center of the ellipse is the midpoint joining the foci $\left( {0, - 2} \right)$ and $\left( {0, - 6} \right)$, 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 $\left( {0, - 4} \right)$, let the required equation of the ellipse be

Since the foci have the coordinates $F\left( {0,ae} \right)$, $F'\left( {0, - ae} \right)$, we have $2ae = FF'$

Using this for the given foci $\left( {0, - 2} \right)$, $\left( {0, - 6} \right)$, we have

It is also given that $2a = 8 \Rightarrow a = 4$. Putting these values in equation (ii), we have

Putting the values of ${a^2}$ and ${b^2}$ in equation (i), we have

This is the required equation of the ellipse.