# 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.