JEE Advanced 2025 · Paper 2, Mathematics question 7

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ 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 $\angle ROM = \dfrac{\pi}{6}$ and $\angle SOM = \dfrac{\pi}{3}$, 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?