# Example of Finding Equation of Ellipse

Example: Find an equation of the ellipse having centre at origin, focus at $\left( {3,0} \right)$ and one vertex at the point $\left( {5,0} \right)$.
Since the focus of an ellipse is at point $\left( {3,0} \right)$, so we take it as $ae = 3$. Since the vertex of an ellipse is at the point $\left( {5,0} \right)$, so by comparing we have $a = 5$.
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 $\left( {0, - 2} \right)$ and $\left( {0, - 6} \right)$, also length of major axis is $8$.
Centre of the ellipse is the midpoint joining the foci $\left( {0, - 2} \right)$ and $\left( {0, - 6} \right)$, so the centre of ellipse can be find using midpoint formula, we have

Since the foci lie on Y-axis with centre $\left( {0, - 4} \right)$, so let the required equation of ellipse will be

Since the foci have the coordinates $F\left( {0,ae} \right)$, $F'\left( {0, - ae} \right)$, so 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 ellipse.