• # question_answer Define the collections $\{{{\text{E}}_{1}},{{\text{E}}_{2}},\,{{\text{E}}_{3}},\,....\}$of ellipse and $\{{{\text{R}}_{1}},\,{{\text{R}}_{2}},\,{{\text{R}}_{3}},\,.....\}$of rectangles as follows: ${{\text{E}}_{1}}\,:\,\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}\,=\,1$ ${{\text{R}}_{1}}\,:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${{\text{E}}_{1}};$ ${{E}_{n}}\ :$ellipse $\frac{{{x}^{2}}}{a_{n}^{2}}+\frac{{{y}^{2}}}{b_{n}^{2}}=\,1$of largest area inscribed in ${{\text{R}}_{n-1}},\,n\,>\,1\,;$ ${{\text{R}}_{n}}\,:$ rectangle of largest area, with sides parallel to the axes inscribed in ${{\text{E}}_{n}},\,n\,>\,1$ Then which of the following options is/are correct? A) The eccentricities of ${{\text{E}}_{18}}$ and ${{\text{E}}_{19}}$ are not equal B) The distance of a focus from the centre in ${{\text{E}}_{9}}$ is $\frac{\sqrt{5}}{32}$ C) $\sum\limits_{n\,=\,1}^{\text{N}}{{}}$$(area\,of\,{{\text{R}}_{n}})\,<24,$ for each positive integer N D) The length of latus rectum of ${{\text{E}}_{9}}$ is $\frac{1}{6}$

 ${{\text{E}}_{1}}\,\Rightarrow \frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1$ $l=6\,\text{cos}\theta$ $\text{b}=4\,\text{sin}\theta$ Area $=12\times \text{sin}2\theta$ ${{\text{A}}_{\text{max}}}=12$ $\text{sin}\,2\theta \,=1$ $2\theta =\frac{\pi }{2}$ $\theta =\frac{\pi }{4}$ ${{\text{E}}_{2}}:a\,=\frac{3}{\sqrt{2}};b=\frac{2}{\sqrt{2}}:$$a=3;r=\frac{1}{\sqrt{2}};\text{b}=2;r=\frac{1}{\sqrt{2}}$ (i) ${{e}^{2}}=1-\frac{{{b}^{2}}}{{{a}^{2}}}$eccentricities of all ellipse will be equal (ii) For ${{\text{E}}_{9}};$$e=\frac{\sqrt{5}}{3}$and $a=3\times {{\left( \frac{1}{\sqrt{2}} \right)}^{8}}$ $\therefore$ distance of focus from centre $=ae=\frac{3}{16}\times \frac{\sqrt{5}}{3}=\frac{\sqrt{5}}{16}$ (iii) sum of area of rectangles $=12+6+3+...$ $\text{A}=\frac{12}{1-\frac{1}{2}}=24$ (iv) $\text{L}\text{.R}.=\frac{2{{b}^{2}}}{a}=\frac{2\times {{\left( 2\times \frac{1}{16} \right)}^{2}}}{2.\frac{1}{16}}=\frac{2\times \frac{1}{64}}{3\times \frac{1}{16}}=\frac{1}{6}.$