The eccentricity of the conic $16x^2 + 7y^2 = 112$ is

The equation of a chord $AB$ of an ellipse $2x^2 + y^2 = 1$ is $x - y + 1 = 0$. If $O$ is the origin,then $\angle AOB =$

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 $9|N_1P|=4|N_1R|$

If the area of the auxiliary circle of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ (where $a > b$) is twice the area of the ellipse,then the eccentricity of the ellipse is

If tangents are drawn to the ellipse $x^2+2y^2=2$,then the locus of the midpoints of the intercepts made by the tangents between the coordinate axes is