Given:
Foci are
![\[\left( 4,\text{ }0 \right)\text{ }\left( -\text{ }4,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3641a6c1618ba504aad2589187832603_l3.png "Rendered by QuickLaTeX.com")
Eccentricity
![\[=\text{ }1/3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-669649c2bf0d731a932dca4f2e205261_l3.png "Rendered by QuickLaTeX.com")
Now let us find the equation to the ellipse.
We know that the equation of the ellipse whose axes are x and y – axis is given as
By using the formula,
Eccentricity:
It is given that foci
![\[=\text{ }\left( 4,\text{ }0 \right)\text{ }\left( -\text{ }4,\text{ }0 \right)\text{ }=>\text{ }foci\text{ }=\text{ }\left( \pm ae,0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-071b8c208227de70a428b525ddf87007_l3.png "Rendered by QuickLaTeX.com")
Where,
![\[ae\text{ }=\text{ }4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-efce89a2a526e7a6aba2e765e97d00a5_l3.png "Rendered by QuickLaTeX.com")
![\[a\left( 1/3 \right)\text{ }=\text{ }4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d815a5dd525890072d914f2b82b94eea_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[a\text{ }=\text{ }12\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-85d76eed895936cd71b968499f3f38ba_l3.png "Rendered by QuickLaTeX.com")
![\[{{a}^{2}}~=\text{ }144\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-25db9f714a44b15bc73afac7ab195f8d_l3.png "Rendered by QuickLaTeX.com")
By substituting the value of
![\[{{a}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-df4561da3d8b3cc6cc17c772a3212739_l3.png "Rendered by QuickLaTeX.com")
we get
![\[{{b}^{2}}~=\text{ }8{{a}^{2}}/9\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f560cafaae6246e9a02b05209c473614_l3.png "Rendered by QuickLaTeX.com")
![\[{{b}^{2}}~=\text{ }8\left( 144 \right)/9\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7e881fbbb8c7587fa5931c34abca12e9_l3.png "Rendered by QuickLaTeX.com")
So,
![\[=\text{ }128\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-27197fe482ea426245b6c0097b6bd6d8_l3.png "Rendered by QuickLaTeX.com")
So the equation of the ellipse can be given as
