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  4. An ellipse having foci at (3,1) and (1,1) passes through the point (1,3). Then find its eccentricity.

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Q. An ellipse having foci at $(3,1)$ and $(1,1)$ passes through the point $(1,3)$. Then find its eccentricity.

Conic Sections

Solution:

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$P S+P S^{\prime}=2 a$
$\sqrt{4+4}+2=2 a$
$2(\sqrt{2}+1)=2 a$
$ \Rightarrow a=\sqrt{2}+1$
$2 a e=\sqrt{4+0} $
$\Rightarrow 2 a e=2$
$ \Rightarrow a e=1$
$e=\frac{1}{\sqrt{2}+1} \times \frac{\sqrt{2}-1}{\sqrt{2}-1}$
$e=\sqrt{2}-1$