Ellipse eccentricity?

Geometry Level 4

If the line \(2px+y\sqrt{1-p^{2}}=1\) always touches the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) \(\forall \ p \in (-1,1) - \{0\}\), find the eccentricity of this ellipse.

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