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 $P:y^{2}=12x$. 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?

Let T₁ and T₂ be two distinct common tangents to the ellipse E:x²/6+y²/3=1 and the parabola P:y²=12x. Suppose that the tangent T₁ touches P and E at the points A₁ and A₂, respectively and the tangent T₂ touches P and E at the points A₄ and A₃, respectively. Then which of the following statements is(are) true? $\text{Ellipse } E: \dfrac{x^{2}}{6} + \dfrac{y^{2}}{3} = 1$ $\text{Parabola } P: y^{2} = 12x$ $\text{To find: area of quadrilateral } A_1A_2A_3A_4$ $\text{and the point where } T_1, T_2 \text{ meet the } x\text{-axis}$ $1.\ \text{tangent to } P \text{ with slope } m$ $2.\ \text{tangent to } E \text{ with slope } m$ $3.\ \text{common condition} \Rightarrow m,\ \text{then the points}$ $y^{2} = 12x = 4ax \Rightarrow a = 3$ $T: y = mx + \dfrac{a}{m} = mx + \dfrac{3}{m}$ $E: \dfrac{x^{2}}{6} + \dfrac{y^{2}}{3} = 1 \Rightarrow a^{2} = 6,\ b^{2} = 3$ $y = mx \pm \sqrt{a^{2}m^{2} + b^{2}} = mx \pm \sqrt{6m^{2} + 3}$ $\dfrac{3}{m} = \pm\sqrt{6m^{2} + 3} \Rightarrow \dfrac{9}{m^{2}} = 6m^{2} + 3$ $6m^{4} + 3m^{2} - 9 = 0 \Rightarrow 2m^{4} + m^{2} - 3 = 0$ $(2m^{2} + 3)(m^{2} - 1) = 0 \Rightarrow m^{2} = 1 \Rightarrow m = \pm 1$ $T_1: y = x + 3, \qquad T_2: y = -x - 3$ $T_1 \cap x\text{-axis}: x = -3; \quad T_2 \cap x\text{-axis}: x = -3$ $\Rightarrow \text{both meet the } x\text{-axis at } (-3, 0)$ $A_1 = (at^{2}, 2at) = (3, 6) \quad (t = 1)$ $A_4 = (3, -6) \quad (t = -1)$ $A_2 = \left(-\dfrac{a^{2}m}{c}, \dfrac{b^{2}}{c}\right) = (-2, 1)$ $A_3 = (-2, -1)$ $A_2A_3 = 2, \quad A_1A_4 = 12, \quad \text{gap} = 5$ $\text{Area} = \dfrac{1}{2}(A_2A_3 + A_1A_4)\times 5 = \dfrac{1}{2}(2+12)\times 5 = 35$ $\boxed{\text{Answer: statements (1) and (3) are true}}$