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If the latus rectum of an ellipse is equal to one half its minor axis, what is the eccentricity of the ellipse?
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The correct answer is:
$\frac{\sqrt{3}}{2}$
Length of latus rectum of an ellipse is $\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}$ where bis
semi minor axis and a is semi-major axis. As given, $\frac{2 b^{2}}{a}=b$
$\Rightarrow 2 \mathrm{~b}=\mathrm{a} \Rightarrow \frac{\mathrm{b}}{\mathrm{a}}=\frac{1}{2}$
Weknow that eccentricity $\mathrm{e}=\sqrt{1-\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}}=\sqrt{1-\frac{1}{4}}=\frac{\sqrt{3}}{2}$
semi minor axis and a is semi-major axis. As given, $\frac{2 b^{2}}{a}=b$
$\Rightarrow 2 \mathrm{~b}=\mathrm{a} \Rightarrow \frac{\mathrm{b}}{\mathrm{a}}=\frac{1}{2}$
Weknow that eccentricity $\mathrm{e}=\sqrt{1-\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}}=\sqrt{1-\frac{1}{4}}=\frac{\sqrt{3}}{2}$
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