Let $(h, k)$ lie on the circle $C: x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to . . . . . . .

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 $N$

The equation of the ellipse whose focus is $(6, 7)$,directrix is $x + y + 2 = 0$,and eccentricity $e = 1/\sqrt{3}$ is:

The number of values of $c$ such that the line $y=4x+c$ touches the curve $\frac{x^{2}}{4}+y^{2}=1$ is

The equation of the ellipse with $x+y+2=0$ as its directrix,one of its focus at $(1,-1)$ and having eccentricity $e = \frac{2}{3}$ is:

Let $A = \{(x, y) : y = mx + 1\}$,$B = \{(x, y) : x^2 + 4y^2 = 1\}$,and $C = \{(\alpha, \beta) : (\alpha, \beta) \in A \text{ and } (\alpha, \beta) \in B \text{ and } \alpha > 0\}$. If set $C$ is a singleton set,then the sum of all possible values of $m$ is: