\(Squeeze \) 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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