The length of the chord of the ellipse $\frac{x^2}{4} + y^2 = 1$ formed on the line $y = x + 1$ is

Find the coordinates of the foci,the vertices,the length of the major axis,the minor axis,the eccentricity,and the length of the latus rectum of the ellipse $36 x^{2}+4 y^{2}=144$.

Find the equation of the ellipse,whose length of the major axis is $20$ and foci are $(0, \pm 5)$.

Define the collections $\{E_1, E_2, E_3, \ldots\}$ of ellipses and $\{R_1, R_2, R_3, \ldots\}$ 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 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 $E_9$ is $\frac{1}{6}$
$(4)$ $\sum_{n=1}^N (\text{area of } R_n) < 24$,for each positive integer $N$

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x + 7y = 50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x - 25 = 0$ be a directrix of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the corresponding focus be $S$. If from $S$,the perpendicular on the $x$-axis passes through $P$,then the length of the latus rectum of $E$ is equal to

The locus of the point of intersection of the perpendicular tangents to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is