Let T₁ and T₂ be two distinct common tangents to the ellipse $\mathrm{E}:\frac{{\mathrm{x}}^2}{6}+\frac{{\mathrm{y}}^2}{3}=1$ and the parabola P: ${\mathrm{y}}^2=12\mathrm{x}$.
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?

Consider the equation of tangent as $\mathrm{y}=\mathrm{mx}+\mathrm{c}$
$\mathrm{y}=\mathrm{mx}+\mathrm{c}$ Tangent to ${\mathrm{y}}^2=12\mathrm{x}\Rightarrow \mathrm{c}=\frac{3}{\mathrm{m}}\mathrm{y}=\mathrm{mx}+\frac{3}{\mathrm{m}}$
It is also tangent to ellipse $\frac{{\mathrm{x}}^2}{6}+\frac{{\mathrm{y}}^2}{3}=1$
$\begin{array}{l}\Rightarrow {\left(\frac{3}{\mathrm{m}}\right)}^2=6\left({\mathrm{m}}^2\right)+3\Rightarrow \frac{9}{{\mathrm{m}}^2}=6{\mathrm{m}}^2+3\\ 6{\mathrm{m}}^4+3{\mathrm{m}}^2-9=0\end{array}$
${\mathrm{m}}^2=1,\frac{-3}{2}$ (Not possible) 
$\mathrm{m}=\pm 1$
$\mathrm{y}=\mathrm{x}+3,\mathrm{y}=-\mathrm{x}-3$ Are the tangents

Point of intersection of tangents is (–3, 0) 
${\mathrm{A}}_1(3,6){\mathrm{A}}_2(-2,1){\mathrm{A}}_3(-2,-1){\mathrm{A}}_4(3,-6)$
Area of quadrilateral ${\mathrm{A}}_1{\mathrm{A}}_2{\mathrm{A}}_3{\mathrm{A}}_4$
$\begin{array}{l}=\frac{1}{2}\left|\begin{array}{ccccc}3& -2& -2& 3& 3\\ 6& 1& -1& -6& 6\end{array}\right|\\ =\frac{1}{2}\left|\right(3+12)+(2+2)+(12+3)+(18+18\left)\right|=\frac{1}{2}\left|70\right|=35 \mathrm{sq}.\mathrm{units}\end{array}$