Question

The eccentricity of the ellipse, if the minor axis is equal to the distance between the foci, is

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

Answer 1

The correct option is C $\frac{1}{\sqrt{2}}$


According to the question, the minor axis is equal to the distance between the foci i.e. 2b= 2ae
$\Rightarrow e=\frac{b}{a}$ ...(1)
Now, $e=\sqrt{1-\frac{b^2}{a^2}}$
$\Rightarrow e=\sqrt{1-e^2}$ [From (1)]
On squaring both sides, we get:
$e^2=1-e^2$
$\Rightarrow 2e^2=1$
$\Rightarrow e^2=\frac{1}{2}$
$\Rightarrow e=\frac{1}{\sqrt{2}}$