The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$. If one of its directrices is $x = -4$,then the equation of the normal to it at $\left(1, \frac{3}{2}\right)$ is

The area of the region enclosed by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is . . . . . . square units. (in $\pi$)

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ is inscribed in a rectangle of dimensions $2a$ and $2b$ respectively. If the angle between the diagonals of the rectangle is $\tan^{-1}(4\sqrt{3})$,then the eccentricity of that ellipse is

The equation of the ellipse with eccentricity $e = \frac{1}{2}$ and foci at $(\pm 1, 0)$ 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 $N$

If the tangents drawn from the point $(\lambda, 3)$ to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ are perpendicular to each other,then $\lambda = ......$