Question

Let $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b.$ Let $E_2$ be another ellipse such that it touches the end points of major axis of $E_1$ and the foci of $E_2$ are the end points of minor axis of $E_1.$ If $E_1$ and $E_2$ have same eccentricities, then its value is:

• A. $\frac{-1+\sqrt{5}}{2}$
• B. $\frac{-1+\sqrt{8}}{2}$
• C. $\frac{-1+\sqrt{3}}{2}$
• D. $\frac{-1+\sqrt{6}}{2}$

Answer 1

Answer: (A)

$e^2=1-\frac{b^2}{a^2}$
$e^2=1-\frac{a^2}{c^2}$
$\Rightarrow \frac{b^2}{a^2}=\frac{a^2}{c^2}$
$\Rightarrow c^2=\frac{a^4}{b^2}$
$\Rightarrow c=\frac{a^2}{b}$
Also $b=ce$
$\Rightarrow c=\frac{b}{e}$
$\frac{b}{e}=\frac{a^2}{b}$
$\Rightarrow e=\frac{b^2}{a^2}=1-e^2$
$\Rightarrow e^2+e-1=0$
$\Rightarrow e=\frac{-1+\sqrt{5}}{2}$