Define the collections {E1, E2, E3,……} of ellipses and {R1, R2, R3,…..} of rectangles as follows:$E_1:\frac{x^2}{9}+\frac{y^2}{4}=1;$

R1: Rectangle of largest area, with sides parallel to the axes, inscribed in E1;$\text{ En: Ellipse }\frac{x^2}{a_n^2}+\frac{y^2}{b_n^2}=1\text{ of largest area inscribed in }{R}_{n-1},n>1\text{ ; }$Rn: Rectangle of largest area, with sides parallel to the axes, inscribed in En, n>1. Then which of the following option(s) is/are correct?

$\text{ Area of }{\mathrm{R}}_1=3\sin 2\theta \text{ ; for this to be maximum }$

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

Hence for subsequent areas of rectangles $R_n$

 to be maximum the coordinates will be in GP with common ratio 

$r=\frac{1}{\sqrt{2}}\Rightarrow a_n=\frac{3}{{\left(\sqrt{2}\right)}^{n-1}};b_n=\frac{3}{{\left(\sqrt{2}\right)}^{n-1}}$

Eccentricity of all the ellipses will be same

          Distance of a focus from the centre in$E_9=a_9{e}_9=\sqrt{a_9^2-b_9^2}=\frac{\sqrt{5}}{16}$

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

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

$\Rightarrow \sum _{n=1}^{\infty }\left(Areaof{R}_n\right)<24$for each positive integer N