Question

The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus-rectum, is

• A. $\frac{\sqrt{5}-1}{2}$
• B. $\frac{\sqrt{5}+1}{2}$
  
• C. $\frac{\sqrt{5}-1}{4}$
  
• D. none of these

Answer 1

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

According to the question, the distance between the foci is equal to the length of the latus rectum
$\frac{2b^2}{a}=2ae$
$\Rightarrow b^2=a^{2}e$
Now, $e=\sqrt{1-\frac{b^2}{a^2}}$
$\Rightarrow e=\sqrt{1-\frac{a^{2}c}{a^2}}$
$\Rightarrow e=\sqrt{1-e}$
On squaring both sides, we get:
$e^2+e-1=0$
$\Rightarrow e=\frac{-1\pm \sqrt{1+4}}{2}$
$\Rightarrow e=\frac{\sqrt{5}-1}{2}$ $(\because ecannotbenegative)$