The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is

Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), y_1<0, y_2<0$, be the end points of the latus rectum of the ellipse $x^2+4 y^2=4$. The equations of parabolas with latus rectum $P Q$ are

$(A)$ $x^2+2 \sqrt{3} y=3+\sqrt{3}$

$(B)$ $x^2-2 \sqrt{3} y=3+\sqrt{3}$

$(C)$ $x^2+2 \sqrt{3} y=3-\sqrt{3}$

$(D)$ $x^2-2 \sqrt{3} y=3-\sqrt{3}$

Suppose that the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ are $\left(f_1, 0\right)$ and $\left(f_2, 0\right)$ where $f_1>0$ and $f_2<0$. Let $P _1$ and $P _2$ be two parabolas with a common vertex at $(0,0)$ and with foci at $\left(f_1, 0\right)$ and $\left(2 f_2, 0\right)$, respectively. Let $T_1$ be a tangent to $P_1$ which passes through $\left(2 f_2, 0\right)$ and $T_2$ be a tangent to $P_2$ which passes through $\left(f_1, 0\right)$. The $m_1$ is the slope of $T_1$ and $m_2$ is the slope of $T_2$, then the value of $\left(\frac{1}{m^2}+m_2^2\right)$ is

Latus rectum of ellipse $4{x^2} + 9{y^2} - 8x - 36y + 4 = 0$ is

Find the equation for the ellipse that satisfies the given conditions: Length of major axis $26$ foci $(±5,\,0)$

Define the collections $\left\{ E _1, E _2, E _3, \ldots ..\right\}$ of ellipses and $\left\{ R _1, K _2, K _3, \ldots ..\right\}$ of rectangles as follows : $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$

$K _1$ : 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 $E_Q$ is $\frac{1}{6}$

$(4)$ $\sum_{n=1}^N\left(\right.$ area of $\left.R_2\right)<24$, for each positive integer $N$