If $ax^2 + by^2 = 15$ is the equation of the ellipse for which the distance between its foci is $2$ and the distance between its directrices is $5$,then $a + b =$

The coordinate of a point on the auxiliary circle of the ellipse $x^{2}+2y^{2}=4$ corresponding to the point on the ellipse whose eccentric angle is $60^{\circ}$ will be

Let $S$ and $S'$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta S'BS$ is a right-angled triangle with the right angle at $B$ and $\text{Area}(\Delta S'BS) = 8 \text{ sq. units}$,then the length of the latus rectum of the ellipse is

If $x \cos \alpha + y \sin \alpha = 4$ is a tangent to $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$,then the value of $\alpha$ 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 $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$.

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 $N$