Let $P$ is any point on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ . $S_1$ and $S_2$ its foci then maximum area of $\Delta PS_1S_2$ is (in square units)

If $PQ$ is a double ordinate of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola. Then the eccentricity $e$ of the hyperbola satisfies

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 $N$

If tangents are drawn to the ellipse $x^2 + 2y^2 = 2$ at all points on the ellipse other than its four vertices than the mid points of the tangents intercepted between the coordinate axes lie on the curve

The value of $\lambda $, for which the line $2x - \frac{8}{3}\lambda y = - 3$ is a normal to the conic ${x^2} + \frac{{{y^2}}}{4} = 1$ is

An ellipse having foci at $(3, 1)$ and $(1, 1) $ passes through the point $(1, 3),$ then its eccentricity is