Area of R₁ = 3sin 2θ, for this to be maximum

$\Rightarrow \mathrm{\theta }=\frac{\mathrm{\pi }}{4}\Rightarrow (\frac{3}{\sqrt{2}},\frac{2}{\sqrt{2}})$                                      

Hence for subsequent areas of rectangles R_n to be maximum the coordinates will be in G.P. with common ratio $\mathrm{r}=\frac{1}{\sqrt{2}}\Rightarrow {\mathrm{a}}_{\mathrm{n}}=\frac{3}{{\left(\sqrt{2}\right)}^{\mathrm{n}-1}};{\mathrm{b}}_{\mathrm{n}}=\frac{3}{{\left(\sqrt{2}\right)}^{\mathrm{n}-1}}$

Eccentricity of all the ellipses will be same

Distance of a focus from the centre in E₉ = a₉e₉ = $\sqrt{{\mathrm{a}}_9^2-{\mathrm{b}}_9^2}=\frac{\sqrt{5}}{16}$

Length of latus rectum of E₉ = $\frac{2{\mathrm{b}}_9^2}{{\mathrm{a}}_9}=\frac{1}{6}$

$\sum _{\mathrm{n}=1}^{\mathrm{\infty }}\mathrm{Area}\mathrm{of}{\mathrm{R}}_{\mathrm{n}}=12+\frac{12}{2}+\frac{12}{4}+.........\mathrm{\infty }=24$

$\sum _{\mathrm{n}=1}^{\mathrm{N}}\left(\mathrm{area}\mathrm{of}{\mathrm{R}}_{\mathrm{n}}\right)$ < 24, for each positive integer N