There are exactly two points on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ whose distance from its centre is same and is equal to $\sqrt{\frac{a^2 + 2b^2}{2}}$. Then the eccentricity of the ellipse is:
Step-by-Step Solution
Key Concept: Equidistant points on an ellipse from its center are located at the vertices of either axis, leading to a constraint on eccentricity.
For exactly two points on the ellipse with equal distance from the center, these points must be endpoints of either the major or minor axis. If they are vertices of the major axis, then $a = \sqrt{\frac{a^2 + 2b^2}{2}}$, giving $a^2 = 2b^2$. This yields $e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{1}{2}} = \frac{1}{\sqrt{2}}$.
Correct Answer: 2