The locus of the mid-points of the chords of an ellipse $x^{2}+4y^{2}=4$ that are drawn from the positive end of the minor axis is

The area (in sq. units) of the triangle formed by the tangent and normal to the ellipse $9x^2 + 4y^2 = 72$ at the point $(2, 3)$ with the $X$-axis is

The centre of an ellipse is $C$,$PN$ is any ordinate,and $A$,$A'$ are the end points of the major axis. Then the value of $\frac{PN^2}{AN \cdot A'N}$ is

For what value of $\lambda$ does the line $y = x + \lambda$ touch the ellipse $9x^2 + 16y^2 = 144$?

The length of the latus rectum of $x^2+3y^2=12$ is

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 $9|N_1P|=4|N_1R|$