Question

A series of concentric ellipses ${ϵ}_1,{ϵ}_2,{ϵ}_3,\cdots {\in }_n$ are drawn such that ${ϵ}_n$ touches the extremities of the major axis of ${ϵ}_{n-1}$ and the focii of ${ϵ}_n$ coincides with the extremities of the minor axis of ${ϵ}_{n-1}$ if the $e$ of the ellipse is independent on $n$, then the value of $e$?

Answer 1

$\begin{array}{c}{b}^{\prime }=a,a^{\prime }e=b\\ b{}^{\prime 2}=a{}^{\prime 2}-a{}^{\prime 2}{e}^2\\ a^2=\frac{b^2}{e^2}-b^2\\ e^2=\frac{b^2}{a^2}\left(1-e^2\right)={\left(1-e^2\right)}^2\\ 1+e^2=\pm e\\ e=\frac{-1\pm \sqrt{5}}{2}\\ e=\frac{\sqrt{5}-1}{2}\end{array}$