Question

Define the collections $\{E_1,E_2,E_3,...\}$ of ellipses and $\{R_1,R_2,R_3,...\}$ of rectangles as follows:
$E_1:\frac{x^2}{9}+\frac{y^2}{4}=1;$
$R_1:$ 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 option is/are correct?

• A. $\sum _{n=1}^N\left(areaof{R}_n\right)<24,$for each positive integers $N$
• B. The eccentricities of $E_{18}$ and $E_{19}$ are NOT equal
• C. The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
• D. The length of latus rectum of $E_9$ is $\frac{1}{6}$

Answer 1

Answer: (A), (D)

The length of latus rectum of $E_9$ is $\frac{1}{6}$


Equation of Eliipse $E_1\Rightarrow \frac{x^2}{9}+\frac{y^2}{4}=1$
$\therefore$ $R_1$ is a rectangle of maximum area inscribed in $E_1$
$l=2\left(3\cos \theta \right)=6\cos \theta$
$b=2\left(2\sin \theta \right)=4\sin \theta$
Area $=12\sin 2\theta$
$A_{max}=12$ when $\sin 2\theta \to 1\Rightarrow \theta \to \frac{\pi }{4}$
$E_n:\frac{x^2}{a_n^2}+\frac{y^2}{b_n^2}=1$ of largest area inscribed in $R_{n-1};n>1$
$E_2:\frac{x^2}{(a_1\cos \frac{\pi }{4}{)}^2}+\frac{y^2}{(b_1\cos \frac{\pi }{4}{)}^2}=1$
$\Rightarrow E_2:\frac{x^2}{(\frac{3}{\sqrt{2}}{)}^2}+\frac{y^2}{(\frac{2}{\sqrt{2}}{)}^2}=1$
$R_2$ is a rectangle of maximum area inscribed in $E_2$
$l=2\left(\frac{3}{\sqrt{2}}\cos \theta \right)=\frac{6}{\sqrt{2}}\cos \theta$
$b=2\left(\frac{2}{\sqrt{2}}\sin \theta \right)=\frac{4}{\sqrt{2}}\sin \theta$
Area $=6\sin 2\theta$
Hence for subsequent area of rectangles $R_n$ to be maximum then coordinates will be in G.P with common ratio $r=\frac{1}{\sqrt{2}}$
$\Rightarrow a_n=\frac{3}{(\sqrt{2}{)}^{n-1}}$ and $b_n=\frac{2}{(\sqrt{2}{)}^{n-1}}$
Eccentricty of all the ellipse will be same i.e.
$e=\sqrt{1-\frac{b^2}{a^2}}\Rightarrow (e{)}_{E_{18}}=(e{)}_{E_{19}}$
Distance of a focus from the centre in $E_9=a_9.e_9=\frac{3}{(\sqrt{2}{)}^8}\times \sqrt{1-\frac{4}{9}}=\frac{\sqrt{5}}{16}$
Length of latus rectum of $E_9=\frac{2(b_9{)}^2}{a_9}=\frac{2(\frac{2}{(\sqrt{2}{)}^8}{)}^2}{\frac{3}{(\sqrt{2}{)}^8}}=\frac{1}{6}$
$\therefore \sum _{n=1}^{\infty }$ Area of $R_n=12+\frac{12}{2}+\frac{12}{2^2}+....=\frac{12}{1-\frac{1}{2}}=24$
$\sum _{n=1}^N\left(Areaof{R}_n\right)<24,$ For each positive integer N.