An ellipse having foci at (3, 1) and (1, 1) passes through point (1, 3), then its eccentricity

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10-2. Parabola, Ellipse, Hyperbola

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An ellipse having foci at $(3, 1)$ and $(1, 1) $ passes through the point $(1, 3),$ then its eccentricity is

A

$\sqrt 2 - 1$

B

$\sqrt 3 - 1$

C

$\frac{1}{2}\left( {\sqrt 2 - 1} \right)$

D

$\frac{1}{2}\left( {\sqrt 3 - 1} \right)$

Solution

$S \equiv(3,1), S^{\prime} \equiv(1,1)$ and $P \equiv(1,3)$

$\Rightarrow P S=\sqrt{(3-1)^{2}+(1-3)^{2}}=2 \sqrt{2}, P S^{\prime}=\sqrt{(1-1)^{2}+(1-3)^{2}}=2$

Using definition of an ellipse $P S+P S^{\prime}=2 a \Rightarrow a=\sqrt{2}+1$

Also $S S^{\prime}=2 a e=2 \Rightarrow e=\frac{1}{a}=\frac{1}{\sqrt{2}+1}=\sqrt{2}-1$

Standard 11

Mathematics

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