There are exactly two points on the ellipse \( \mathrm{P}^{16: 4} \...

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There are exactly two points on the ellipse
\( \mathrm{P}^{16: 4} \) \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) whose distances from its center are the

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same and are equal to \( \sqrt{\left(a^{2}+2 b^{2}\right) / 2} \). Then the eccentricity of the ellipse is:
(1) \( \frac{1}{2} \)
(2) \( \frac{1}{\sqrt{2}} \)
(3) \( \frac{1}{3} \)
(4) \( \frac{1}{3 \sqrt{2}} \)
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