If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is : [2024]

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Solution

Q.
If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is : [2024]

1 $\frac{\sqrt{5}}{3}$

2 $\frac{2}{\sqrt{5}}$

3 $\frac{\sqrt{3}}{2}$

4 $\frac{1}{\sqrt{3}}$

Ans.
(2)

Let $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ be the given ellipse

We have, minor axis $\frac{1}{2} (2 a e)$ , where e is eccentricity

$\Rightarrow 2 b = a e \Rightarrow 4 b^{2} = a^{2} e^{2}$

$\Rightarrow 4 b^{2} = a^{2} - b^{2}$ [ $∵ b^{2} = a^{2} - a^{2} e^{2}$ ]

$\Rightarrow 5 b^{2} = a^{2}$

Now, $e = \sqrt{1 - \frac{b^{2}}{a^{2}}} = \sqrt{1 - \frac{1}{5}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}}$

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