If lines $3x + 2y = 10$ and $-3x + 2y = 10$ are tangents at the extremities of the latus rectum of an ellipse whose centre is the origin,then the length of the latus rectum of the ellipse is:

For $\alpha$ belonging to an interval of length $\beta$,suppose $(\alpha, -\alpha)$ is an interior point of the ellipse $4x^2 + 5y^2 = 1$. Then,$(6\beta - 4)^{201} + 201 = $

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ such that $y_1 > 0$ and $y_2 > 0$. Let $C$ denote the circle $x^2+y^2=9$,and $M$ be the point $(3,0)$. Suppose the line $x=x_1$ intersects $C$ at $R$,and the line $x=x_2$ intersects $C$ at $S$,such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle ROM = \frac{\pi}{6}$ and $\angle SOM = \frac{\pi}{3}$,where $O$ denotes the origin $(0,0)$. Let $|XY|$ denote the length of the line segment $XY$. Then which of the following statements is (are) True?
$(A)$ The equation of the line joining $P$ and $Q$ is $2x+3y=3(1+\sqrt{3})$
$(B)$ The equation of the line joining $P$ and $Q$ is $2x+y=3(1+\sqrt{3})$
$(C)$ If $N_2=(x_2, 0)$,then $3|N_2Q|=2|N_2S|$
$(D)$ If $N_1=(x_1, 0)$,then $9|N_1P|=4|N_1R|$

Let $x = 9$ be a directrix of an ellipse $E$, whose centre is at the origin and eccentricity is $1/3$. Let $P(\alpha, 0), \alpha > 0$, be a focus of $E$ and $AB$ be a chord passing through $P$. Then the locus of the mid point of $AB$ is :

The latus rectum of an ellipse is $10$ and the minor axis is equal to the distance between the foci. The equation of the ellipse is

Let $E$ be an ellipse whose major axis is the $X$-axis and minor axis is the $Y$-axis. If the distances of a point $P \left(\frac{5}{2}, 2 \sqrt{3}\right)$ on $E$ from its foci are $\frac{7}{2}$ and $\frac{13}{2}$,then the eccentricity of the ellipse $E$ is