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Define the collections $\left\{ E _1, E _2, E _3, \ldots ..\right\}$ of ellipses and $\left\{ R _1, K _2, K _3, \ldots ..\right\}$ of rectangles as follows : $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$
$K _1$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _1$;
$E_n$ : ellipse $\frac{x^2}{a_n^2}+\frac{y^2}{b_{n}^2}=1$ of largest area inscribed in $R_{n-1}, n>1$;
$R _{ n }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _{ n }, n >1$.
Then which of the following options is/are correct?
$(1)$ The eccentricities of $E _{18}$ and $E _{19}$ are NOT equal
$(2)$ The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
$(3)$ The length of latus rectum of $E_Q$ is $\frac{1}{6}$
$(4)$ $\sum_{n=1}^N\left(\right.$ area of $\left.R_2\right)<24$, for each positive integer $N$
$1,2$
$1,3$
$1,4$
$3,4$
Solution
Area of $R_1=3 \sin 2 \theta$; for this to be maximum
$\Rightarrow \theta=\frac{\pi}{4}$ $\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 GP with common ratio $r =\frac{1}{\sqrt{2}} \Rightarrow a _{ n }=\frac{3}{(\sqrt{2})^{ n -1}} ; b _{ n }=\frac{3}{(\sqrt{2})^{ n -1}}$
Eccentricity of all the ellipses will be same
Distance of a focus from the centre in $E_9=a_9 e_9=\sqrt{a_g^2-b_9^2}=\frac{\sqrt{3}}{16}$
Length of latus rectum of $E_9=\frac{2 b_9^2}{a_9}=\frac{1}{6}$
$\because \sum_{n=1}^{\infty} \text { Area of } R_n=12+\frac{12}{2}+\frac{12}{4}+\ldots . \infty=24$
$\Rightarrow \sum_{ n =1}^{ N }\left(\text { area of } R _{ n }\right)<24, $for each positive integer $N$
