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 $\angle R O M=\frac{\pi}{6}$ and $\angle S O M=\frac{\pi}{3}$, where $O$ denotes the origin $(0,0)$. Let $|X Y|$ denote the length of the line segment $X Y$.

Then which of the following statements is (are) TRUE?

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

Define the collections {E₁, E₂, E₃, ...} of ellipses and {R₁, R₂, R₃.....} of rectangles as follows :

$${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$

R₁ : rectangle of largest area, with sides parallel to the axes, inscribed in E₁;

En : ellipse $${{{x^2}} \over {a_n^2}} + {{{y^2}} \over {b_n^2}} = 1$$ of the largest area inscribed in $${R_{n - 1}},n > 1$$;

R_n : rectangle of largest area, with sides parallel to the axes, inscribed in E_n, n > 1.

Then which of the following options is/are correct?

Consider two straight lines, each of which is tangent to both the circle x² + y² = (1/2) and the parabola y² = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is $$\sqrt 2 $$, then which of the following statement(s) is (are) TRUE?