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  4. Eccentricity of the ellipse whose latusrectum is equal to the distance between two focus points, is

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Q. Eccentricity of the ellipse whose latusrectum is equal to the distance between two focus points, is

Conic Sections

Solution:

According to given condition, $\frac{2 b^2}{a}=2 a e$
$\Rightarrow b^2=a^2 e $ or $e=\frac{b^2}{a^2}$
Also, $e=\sqrt{1-\frac{b^2}{a^2}}$ or $e^2=1-e$
or $ e^2+e-1=0$
Therefore, $e=\frac{-1 \pm \sqrt{5}}{2}$
$As, e<1$
$\therefore e=\frac{\sqrt{5}-1}{2}$