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.