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  4. The length of the semi-latus rectum of an ellipse is one third of its major axis, its eccentricity would be

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Q. The length of the semi-latus rectum of an ellipse is one third of its major axis, its eccentricity would be

Conic Sections

Solution:

Let eq. of ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$,
length $=\frac{b^{2}}{a}=\frac{a^{2}\left(1-e^{2}\right)}{a}=a\left(1-e^{2}\right)$
Given $a\left(1-e^{2}\right)=\frac{1}{3}(2 a)$
$\Rightarrow 1-e^{2}=\frac{2}{3} $
$\Rightarrow e^{2}=1-\frac{2}{3}=\frac{1}{3} $
$\Rightarrow e=\frac{1}{\sqrt{3}}$