(i)
![\[Eccentricity\text{ }e\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}~\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7b0c60670cfdbfb7784afffddde6422b_l3.png "Rendered by QuickLaTeX.com")
and
![\[foci\text{ }\left( \pm \text{ }2,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ee40424acfb5d70fb76e81f7d071fcf9_l3.png "Rendered by QuickLaTeX.com")
Given:
Eccentricity
![\[e\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a85b380d37e0d43f824d158e47eac774_l3.png "Rendered by QuickLaTeX.com")
![\[Foci\text{ }\left( \pm \text{ }2,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8527d9f79cb8326c4a6fc982297aef55_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:
![\[{{b}^{2}}~=\text{ }3{{a}^{2}}/4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-095595bc5fe3b94f14afc64ebf97a24a_l3.png "Rendered by QuickLaTeX.com")
It is given that foci
![\[\left( \pm \text{ }2,\text{ }0 \right)\text{ }=>foci\text{ }=\text{ }\left( \pm ae,\text{ }0 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b4e7f0dfe080ff99c15b18950310ed3e_l3.png "Rendered by QuickLaTeX.com")
Where,
![\[ae\text{ }=\text{ }2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-36db174512e832843bb83deff84b84bf_l3.png "Rendered by QuickLaTeX.com")
![\[a\left( 1/2 \right)\text{ }=\text{ }2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9725eb3795a6fa002a841b6dbc34c4fb_l3.png "Rendered by QuickLaTeX.com")
Or,
![\[a\text{ }=\text{ }4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ac1222b81394580973acc531cc4c93c9_l3.png "Rendered by QuickLaTeX.com")
![\[{{a}^{2}}~=\text{ }16\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-657e7f7a5c79f1a7da0b18416810e4ff_l3.png "Rendered by QuickLaTeX.com")
We know
![\[{{b}^{2}}~=\text{ }3\left( 16 \right)/4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b3f6f78b04a4d0bfa37f5bc792b5f195_l3.png "Rendered by QuickLaTeX.com")
So,
![\[=\text{ }12\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dd99338e6f3e8f7cea5653cc0ec32a13_l3.png "Rendered by QuickLaTeX.com")
So the equation of the ellipse can be given as
![\[3{{x}^{2}}~+\text{ }4{{y}^{2}}~=\text{ }48\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-62a7b1e5d651850ba7dc9f718011da78_l3.png "Rendered by QuickLaTeX.com")
∴ The equation of the ellipse is
(ii) eccentricity
![\[e\text{ }=\text{ }2/3~\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-670309e485edc0d511613bb12e202e71_l3.png "Rendered by QuickLaTeX.com")
and length of latus rectum
![\[=\text{ }5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e45ddd13b94c34a71f1a55ef7d8c9ef2_l3.png "Rendered by QuickLaTeX.com")
Given:
Eccentricity
![\[e\text{ }=\text{ }2/3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-af1db0aea52216b91e881fe0196c888f_l3.png "Rendered by QuickLaTeX.com")
Length of latus – rectum
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:
By using the formula, length of the latus rectum is
![\[2{{b}^{2}}/a\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a3ecfe1d612a1e0e995bd1475f8f34e4_l3.png "Rendered by QuickLaTeX.com")
So the equation of the ellipse can be given as
![\[20{{x}^{2}}~+\text{ }36{{y}^{2}}~=\text{ }405\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-26aa560fa51c56211b100b042a2d4758_l3.png "Rendered by QuickLaTeX.com")
∴ The equation of the ellipse is