\(x^2+y^2+xy+2=3\left(x+y\right)\)
\(\Leftrightarrow4x^2+4y^2+4xy+8-12x-12y=0\)
\(\Leftrightarrow\left(4x^2+y^2+9+4xy-12x-6y\right)+3y^2-6y+3=4\)
\(\Leftrightarrow\left(2x+y-3\right)^2+\left(\sqrt{3}y-\sqrt{3}\right)^2=4\)
\(\Leftrightarrow\left(\dfrac{2x+y-3}{2}\right)^2+\left(\dfrac{\sqrt{3}y-\sqrt{3}}{2}\right)^2=1\)
Đặt \(\dfrac{2x+y-3}{2}=sina\Rightarrow\dfrac{\sqrt{3}y-\sqrt{3}}{2}=cosa\)
\(\Rightarrow\left\{{}\begin{matrix}y=\dfrac{2}{\sqrt{3}}cosa+1\\x=sina-\dfrac{1}{\sqrt{3}}cosa+1\end{matrix}\right.\)
\(P=\dfrac{3\left(sina-\dfrac{1}{\sqrt{3}}cosa+1\right)+2\left(\dfrac{2}{\sqrt{3}}cosa+1\right)+1}{sina+\dfrac{1}{\sqrt{3}}cosa+8}=\dfrac{3sina+\dfrac{1}{\sqrt{3}}cosa+6}{sina+\dfrac{1}{\sqrt{3}}cosa+8}\)
\(\Rightarrow P.sina+\dfrac{P}{\sqrt{3}}cosa+8P=3sina+\dfrac{1}{\sqrt{3}}cosa+6\)
\(\Rightarrow\left(P-3\right)sina+\left(\dfrac{P-1}{\sqrt{3}}\right)cosa=6-8P\)
\(\Rightarrow\left(P-3\right)^2+\dfrac{\left(P-1\right)^2}{3}\ge\left(6-8P\right)^2\)
\(\Rightarrow47P^2-67P+20\le0\)
\(\Rightarrow\dfrac{20}{47}\le P\le1\)
