Question

A series of concentric ellipses E₁, E₂, ................, E_n are drawn such that E_n touches the extremities of the major axis of _En-1 and the foci of E_n coincide with the extremities of minor axis of _En-1. If the eccentricity of the ellipses is independent of n, then the value of the eccentricity is

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

Answer 1

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

The figure shows two ellipses $E_{n-1}$ and $E_n.$ The eccentricity is given to be indipendent of $n$, implies the ratio of minor axis to the major axis,is same for all the ellipses.


For ellipse $E_{n-1}$, let minor-axis $=b$, major-axis$=a.$
For ellipse $E_n$, we have minor-axis$=a,$ major-axis $=\frac{OB}{e}=\frac{b}{e}$
$[B$ is the focus of $E_n]$
Assuming $e$ to be the eccentricity. Thus, we have $\frac{b}{a}=\frac{a}{b/e}$
i.e. $e=\frac{b^2}{a^2}=1-e^2$
i.e. $e^2+e-1=0$
i.e. $e=\frac{\sqrt{5}-1}{2}$ $[e\text{must be}+ve]$