10 tangents, 20 feets!

Geometry Level 3

Let \(P_i\) and \(P'_i\) be the feet of perpendiculars drawn from foci \(S\) and \(S'\) on a tangent \(T_i\) to the ellipse \(E: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) whose length of major axis is 40 units.

If \( \ \displaystyle \sum_{i=1}^{10} (SP_i)(SP'_i) = 2560\), then find the value of \(100e\).


'\(e\)' denotes the eccentricity of the ellipse.


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