Question

If the length of the minor axis of an ellipse is equal to the distance between their foci, then eccentricity of the ellipse is:

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

Answer 1

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

Length of minor axis of an ellipse $=2b$

Distance between foci $=2ae$

According to the question,

$\Rightarrow$ $2ae=2b$

$\Rightarrow$ $ae=b$

$\Rightarrow$ $a^2{e}^2=b^2$

We know, $b^2=a^2(1-e^2)$

$\Rightarrow$ $a^2{e}^2=a^2(1-e^2)$

$\Rightarrow$ $a^2{e}^2=a^2-a^2{e}^2$

$\Rightarrow$ $2a^2{e}^2=a^2$

$\Rightarrow$ $2e^2=1$

$\Rightarrow$ $e^2=\frac{1}{2}$

$\therefore$ $e=\frac{1}{\sqrt{2}}$