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$K _1$ : 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 $N$
$L _1: x \sqrt{2}+ y -1=0$ and $L _2: x \sqrt{2}- y +1=0$
For a fixed constant $\lambda$, let $C$ be the locus of a point $P$ such that the product of the distance of $P$ from $L_1$ and the distance of $P$ from $L_2$ is $\lambda^2$. The line $y=2 x+1$ meets $C$ at two points $R$ and $S$, where the distance between $R$ and $S$ is $\sqrt{270}$.
Let the perpendicular bisector of $RS$ meet $C$ at two distinct points $R ^{\prime}$ and $S ^{\prime}$. Let $D$ be the square of the distance between $R ^{\prime}$ and $S ^{\prime}$.
($1$) The value of $\lambda^2$ is
($2$) The value of $D$ is