Let $F_1\left(x_1, 0\right)$ and $F_2\left(x_2, 0\right)$, for $x_1<0$ and $x_2>0$, be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1$. Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.

($1$)The orthocentre of the triangle $F_1 M N$ is

($A$) $\left(-\frac{9}{10}, 0\right)$   ($B$) $\left(\frac{2}{3}, 0\right)$    ($C$) $\left(\frac{9}{10}, 0\right)$    ($D$) $\left(\frac{2}{3}, \sqrt{6}\right)$

($2$) If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $M Q R$ to area of the quadrilateral $M F_{\mathrm{I}} N F_2$ is

($A$) $3: 4$     ($B$) $4: 5$     ($C$) $5: 8$     ($D$) $2: 3$

Givan the answer qestion ($1$) and ($2$)

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $\frac{x^{2}}{100}+\frac{y^{2}}{400}=1$.

Find the equation for the ellipse that satisfies the given conditions: Foci $(\pm 3,\,0),\,\, a=4$

If $F_1$ and $F_2$ be the feet of the perpendicular from the foci $S_1$ and $S_2$ of an ellipse $\frac{{{x^2}}}{5} + \frac{{{y^2}}}{3} = 1$ on the tangent at any point $P$ on the ellipse, then $(S_1 F_1) (S_2 F_2)$ is equal to

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 $N$