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

{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, so the required equation of 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 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

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


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

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