Question

A series of concentric ellipses $E_1,E_2,\dots ,E_n$ are drawn such that $E_n$ touches the extremities of the major axis of $E_{n-1}$ and the foci of $E_n$ coincide with the extemities of minor axis of $E_{n-1}.$ If the eccentricites of the ellipse are independent of $n,$ then the value of the eccentricity, is

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

Answer 1

The correct option is B $\frac{\sqrt{5}-1}{2}$

Let $E_n:\frac{x^2}{a_n^2}+\frac{y^2}{b_n^2}=1$
and $E_{n-1}:\frac{x^2}{a_{n-1}^2}+\frac{y^2}{b_{n-1}^2}=1$

Now from given conditions and figure
$a_n=a_{n-1},b_{n}e=b_{n-1}$
$e=\sqrt{1-\frac{a_n^2}{b_n^2}}=\sqrt{1-\frac{b_{n-1}^2}{a_{n-1}^2}}\Rightarrow \frac{a_{n-1}^2{e}^2}{b_{n-1}^2}=\frac{b_{n-1}^2}{a_{n-1}^2}\Rightarrow e^2=(1-e^2{)}^2$
$\Rightarrow e=\pm (1-e^2)$
$\Rightarrow e^2+e-1=0[\because e<1]$
$\Rightarrow e=\frac{\sqrt{5}-1}{2}$
$[\because e>0]$