1. Use the given angles on the circle to find $x_1,x_2$

The circle is $x^2+y^2=9,$ so it has radius $3$ and center at $O=(0,0)$.

Also, $M=(3,0)$ lies on the positive $x$-axis, so $\angle ROM$ is just the polar angle of point $R$.

Since $R$ lies on the circle and has positive $y$-coordinate, $R=(3\cos\tfrac{\pi}{6},\,3\sin\tfrac{\pi}{6})=\left(\frac{3\sqrt3}{2},\frac32\right).$ Because $R$ lies on the line $x=x_1$, we get $x_1=\frac{3\sqrt3}{2}.$

Similarly, $S=(3\cos\tfrac{\pi}{3},\,3\sin\tfrac{\pi}{3})=\left(\frac32,\frac{3\sqrt3}{2}\right),$ so $x_2=\frac32.$

2. Find points $P$ and $Q$ on the ellipse

The ellipse is $\frac{x^2}{9}+\frac{y^2}{4}=1,$ with $y_1>0,y_2>0$.

For $P(x_1,y_1)$:

$x_1=\frac{3\sqrt3}{2} \implies \frac{x_1^2}{9}=\frac{\frac{27}{4}}{9}=\frac34.$ Hence $\frac{y_1^2}{4}=1-\frac34=\frac14 \implies y_1^2=1.$ Since $y_1>0$, $y_1=1.$ Therefore, $P\left(\frac{3\sqrt3}{2},1\right).$

For $Q(x_2,y_2)$:

$x_2=\frac32 \implies \frac{x_2^2}{9}=\frac{\frac94}{9}=\frac14.$ Hence $\frac{y_2^2}{4}=1-\frac14=\frac34 \implies y_2^2=3.$ Since $y_2>0$, $y_2=\sqrt3.$ Therefore, $Q\left(\frac32,\sqrt3\right).$

3. Equation of line joining $P$ and $Q$

Slope: $m=\frac{\sqrt3-1}{\frac32-\frac{3\sqrt3}{2}}=\frac{2(\sqrt3-1)}{3(1-\sqrt3)}=-\frac23.$

So the line is of form $y=-\frac23x+c.$ Using point $Q\left(\frac32,\sqrt3\right)$, $\sqrt3=-\frac23\cdot \frac32+c=-1+c,$ so $c=1+\sqrt3.$ Thus, $y=-\frac23x+(1+\sqrt3).$ Multiply by $3$: $2x+3y=3(1+\sqrt3).$

So Option A is true and Option B is false.

4. Check Option C

Let $N_2=(x_2,0)=\left(\frac32,0\right).$ Since $Q=(x_2,y_2)$ and $S$ lies on the same vertical line $x=x_2$, $|N_2Q|=y_2=\sqrt3,$ $|N_2S|=\frac{3\sqrt3}{2}.$ Now, $3|N_2Q|=3\sqrt3,$ $2|N_2S|=2\cdot \frac{3\sqrt3}{2}=3\sqrt3.$ Hence, $3|N_2Q|=2|N_2S|.$ So Option C is true.

5. Check Option D

Let $N_1=(x_1,0)=\left(\frac{3\sqrt3}{2},0\right).$ Then $|N_1P|=y_1=1,$ $|N_1R|=\frac32.$ So, $9|N_1P|=9,$ $4|N_1R|=4\cdot \frac32=6.$ These are not equal.

So Option D is false.

6. Final conclusion

The true statements are: $\boxed{A,\ C}$

This matches the stored correct answer.