The eccentric angle of the end point of the latus rectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is:

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ of eccentricity $e=\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$ is:

Let each of the two ellipses $E_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b)$ and $E_2: \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, (A < B)$ have eccentricity $\frac{4}{5}$. Let the lengths of the latus recta of $E_1$ and $E_2$ be $\ell_1$ and $\ell_2$, respectively, such that $2\ell_1^2 = 9\ell_2$. If the distance between the foci of $E_1$ is $8$, then the distance between the foci of $E_2$ is:

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 $N$

Find the equations of the tangents to the ellipse $3x^{2} + 4y^{2} = 12$ which are perpendicular to the line $y + 2x = 4$.

$A$ ray of light passing through $(2, 1)$ is reflected at a point $P$ on the $y$-axis and then passes through the point $(5, 3)$. If this reflected ray is the directrix of an ellipse with eccentricity $e = \frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}}$,then the equation of the other directrix can be: