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 $b=c e$
$\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 $
$\Rightarrow e=\frac{-1+\sqrt{5}}{2}$