Question

An ellipse with its minor and major axis parallel to the coordinate axes passes through $(0,0),(1,0)$ and $(0,2)$. One of its foci lies on the y-axis. The eccentricity of the ellipse is?

• A. $\sqrt{3}-1$
• B. $\sqrt{5}-2$
• C. $\sqrt{2}-1$
• D. $\frac{\sqrt{3}-1}{2}$

Answer 1

Answer: (C)

$\sqrt{2}-1$

From the figure,

$(\frac{1}{2},1)$ is the centre of ellipse and $(0,1)$ is one focus of ellipse.

$\therefore$ we have $ae=\frac{1}{2}$, where $e$ is the eccentricity of ellipse.

and $\frac{2b^2}{a}=2\Rightarrow b^2=a$

$(ae{)}^2=a^2-b^2\Rightarrow \frac{1}{4}=a^2-a\Rightarrow 4a^2-4a-1=0$

$a=\frac{4\pm \sqrt{16+16}}{8}=\frac{1\pm \sqrt{2}}{2}$

but $a>0\Rightarrow a=\frac{1+\sqrt{2}}{2}$

$\Rightarrow ae=\frac{1}{2}\Rightarrow e=\sqrt{2}-1$