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?

Let P(x₁, y₁) and Q(x₂, y₂) be two distinct points on the ellipse x²/9 + y²/4 = 1 such that y₁ > 0 and y₂ > 0. Let C denote the circle x² + y² = 9, and M be the point (3,0). Suppose the line x = x₁ intersects C at R, and the line x = x₂ intersects C at S, such that the y-coordinates of R and S are positive. Let ∠ ROM = π/6 and ∠ SOM = π/3, where O denotes the origin (0, 0). Let |XY| denote the length of the line segment XY. Then which of the followin

$\text{Ellipse: } \dfrac{x^2}{9} + \dfrac{y^2}{4} = 1,\quad P(x_1,y_1),\ Q(x_2,y_2),\ y_1,y_2>0$

$C:\ x^2 + y^2 = 9,\quad M(3,0),\quad O(0,0)$

∠ ROM = π/6, ∠ SOM = π/3

$\text{To find: which statements } (1)\text{–}(4) \text{ are TRUE}$

$R = \left(3\cos\dfrac{\pi}{6},\ 3\sin\dfrac{\pi}{6}\right) = \left(\dfrac{3\sqrt3}{2},\ \dfrac{3}{2}\right)$

$S = \left(3\cos\dfrac{\pi}{3},\ 3\sin\dfrac{\pi}{3}\right) = \left(\dfrac{3}{2},\ \dfrac{3\sqrt3}{2}\right)$

$x_1 = \dfrac{3\sqrt3}{2},\qquad x_2 = \dfrac{3}{2}$

$\dfrac{x_1^2}{9} = \dfrac34 \Rightarrow y_1^2 = 4\left(1 - \dfrac34\right) = 1 \Rightarrow y_1 = 1$

$P = \left(\dfrac{3\sqrt3}{2},\ 1\right)$

$\dfrac{x_2^2}{9} = \dfrac14 \Rightarrow y_2^2 = 4\left(1 - \dfrac14\right) = 3 \Rightarrow y_2 = \sqrt3$

$Q = \left(\dfrac{3}{2},\ \sqrt3\right)$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\sqrt3 - 1}{\tfrac{3}{2} - \tfrac{3\sqrt3}{2}} = -\dfrac{2}{3}$

$y - 1 = -\dfrac23\left(x - \dfrac{3\sqrt3}{2}\right)$

$\Rightarrow\ 2x + 3y = 3 + 3\sqrt3 = 3(1 + \sqrt3)$

$(1)\ 2x + 3y = 3(1+\sqrt3)\ \checkmark$

$(2)\ 2x + y = 3(1+\sqrt3)\ \times$

$N_2\left(\dfrac32, 0\right):\ |N_2Q| = \sqrt3,\quad |N_2S| = \dfrac{3\sqrt3}{2}$

$3|N_2Q| = 3\sqrt3 = 2|N_2S| \ \checkmark$

$N_1\left(\dfrac{3\sqrt3}{2}, 0\right):\ |N_1P| = 1,\quad |N_1R| = \dfrac32$

$9|N_1P| = 9 \ \ne\ 6 = 4|N_1R| \ \times$

$\boxed{PQ:\ 2x + 3y = 3(1 + \sqrt3)}$

$\boxed{\text{Correct statements: } (1)\ \text{and}\ (3)}$