Question

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

Options

• \[\frac{\sqrt{3}}{2}\]

   

• \[\frac{2}{\sqrt{3}}\]

   

• \[\frac{1}{\sqrt{2}}\]

   

• \[\frac{\sqrt{2}}{3}\]

Answer 1

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