Question

An ellipse having foci at $(3,1)$ and $(1,1)$ passes through the point $(1,3)$. Its eccentricity is:

• A. $\sqrt{2}-1$
• B. $\sqrt{3}-1$
• C. $\frac{1}{2}(\sqrt{2}-1)$
• D. $\frac{1}{2}(\sqrt{3}-1)$

Answer 1

The correct option is A $\sqrt{2}-1$

$S\equiv (3,1),S^{\prime }\equiv (1,1)$ and $P\equiv (1,3)$
$\Rightarrow PS=\sqrt{(3-1{)}^2+(1-3{)}^2}=2\sqrt{2},P{S}^{\prime }=\sqrt{(1-1{)}^2+(1-3{)}^2}=2$
Using definition of an ellipse $PS+P{S}^{\prime }=2a\Rightarrow a=\sqrt{2}+1$
Also $S{S}^{\prime }=2ae=2\Rightarrow e=\frac{1}{a}=\frac{1}{\sqrt{2}+1}=\sqrt{2}-1$