# 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

${a^2} – {b^2} = {a^2}{e^2}$
$\begin{gathered} \Rightarrow {b^2} = {a^2} – {\left( {ae} \right)^2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( 5 \right)^2} – {\left( 3 \right)^2} = 25 – 9 = 16 \\ \Rightarrow b\,\,\,\, = \pm 4 \\ \end{gathered}$

Since the focus lies on the X-axis, the required equation of the ellipse is
$\begin{gathered} \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \\ \Rightarrow \frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1 \\ \end{gathered}$

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

$\left( {\frac{{0 + 0}}{2},\frac{{ – 2 – 6}}{2}} \right) = \left( {0, – 4} \right)$

Since the foci lie on the Y-axis with center $\left( {0, – 4} \right)$, let the required equation of the ellipse be
$\begin{gathered} \frac{{{{\left( {y – \left( { – 4} \right)} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {x – 0} \right)}^2}}}{{{b^2}}} = 1 \\ \Rightarrow \frac{{{{\left( {y + 4} \right)}^2}}}{{{a^2}}} + \frac{{{x^2}}}{{{b^2}}} = 1\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ {a^2} – {b^2} = {a^2}{e^2}\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right) \\ \end{gathered}$

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

$\begin{gathered} 2ae = \left| { – 6 – \left( { – 2} \right)} \right| = \left| { – 6 + 2} \right| = \left| { – 4} \right| = 4 \\ \Rightarrow ae = 2 \\ \end{gathered}$

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

$\begin{gathered} {4^2} – {b^2} = {2^2} \\ \Rightarrow {b^2} = 16 – 4 = 12 \\ \end{gathered}$

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

$\Rightarrow \frac{{{{\left( {y + 4} \right)}^2}}}{{16}} + \frac{{{x^2}}}{{12}} = 1$

This is the required equation of the ellipse.