Let P_i and P_i’ be the feet of the perpendiculars drawn from foci S, S’ on a tangent T_i to an ellipse whose length of semi-major axis is 20, if $\sum _{\mathrm{i}=1}^{10} \left({\mathrm{SP}}_{\mathrm{i}}\right)\left({\mathrm{S}}^{\mathrm{\prime }}{\mathrm{P}}_{\mathrm{i}}^{\mathrm{\prime }}\right)=2560$ then the value of eccentricity is

$$\begin{gathered} \text{From the given data } \\ We know that \text{ SP.S'P' }={\mathrm{b}}^2\text{; } \\ \begin{array}{r}\therefore \sum _{\mathrm{i}=1}^{10} \left({\mathrm{SP}}_{\mathrm{i}}\right)\left({\mathrm{S}}^{\mathrm{\prime }}{\mathrm{P}}_{\mathrm{i}}^{\mathrm{\prime }}\right)==2560 \\ \Rightarrow 10{\mathrm{b}}^2= 2560 \\ \Rightarrow {\mathrm{b}}^2=256 and given \mathrm{a}=20 \\ Now e=\sqrt{\frac{a^2-b^2}{a^2}} \\ \Rightarrow \mathrm{e}=\frac{3}{5} \\ \end{array} \end{gathered}$$