$$Squeeze$$ Ellipse and Calculus $$\Longrightarrow Ellipsulus$$

Calculus Level 5

Line segments $$AK_{1},AK_{2},......,AK_{n}$$ Are drawn from A(1,1) where $$K_{1},K_{2},....,K_{n}$$ are points in first quadrant on $$\frac{(x-1)^{2}}{a^{2}}+\frac{(y-1)}{b^{2}})$$

(a>b).such that the chord $$AK_{r}$$ makes an angle of $$\theta=\frac{r\pi}{2n}$$ with the positive x axis.

$\displaystyle{lim_{n\rightarrow\infty}(\frac{1}{n}(\sum_{r=1}^{n}(AK_{r})^{(lim_{n\rightarrow\infty}\sum_{k=1}^{n}(\frac{k}{n^{2}+n+2k}))^{-c}}))=\frac{S}{d} }$

S is the area of the ellipse

M and M' are the feet of perpendiculars from the foci S and S' to the tangents at any point P on the ellipse . If $$SM=2S'M'$$ $$S'P=2\pi$$ d is the length of SP

find the value of $$2c$$

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