Mathematics - Ellipse Question with Solution | TestHub

Let $P\left(x_1,y_1\right)$ and $Q\left(x_2,y_2\right)$ 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?