Given:
We need to find the eccentricity of an ellipse.
(i) If latus – rectum is half of its minor axis
We know that the length of the semi – minor axis is
![\[b\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-395b0d1850e0b5677f132c9666f02418_l3.png "Rendered by QuickLaTeX.com")
and the 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")
![\[2{{b}^{2}}/a\text{ }=\text{ }b\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d6b296e7f2e72488bd22b9c06463c45e_l3.png "Rendered by QuickLaTeX.com")
And,
![\[a\text{ }=\text{ }2b\text{ }\ldots .\text{ }\left( 1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-43187320b40cb74adec3f365987c28ea_l3.png "Rendered by QuickLaTeX.com")
By using the formula,
We know that eccentricity of an ellipse is given as
(ii) If latus – rectum is half of its major axis
We know that the length of the semi – major axis is a and the length of the latus – rectum is
So,
![\[{{a}^{2}}~=\text{ }2{{b}^{2}}~\ldots .\text{ }\left( 1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-960768e43f2ecebf7975f9e31df7abba_l3.png "Rendered by QuickLaTeX.com")
By using the formula,
We know that eccentricity of an ellipse is given as
