\(\left(d\right)\cap\left(Ox\right)\Leftrightarrow\left(m-2\right)x+m-3=0\Leftrightarrow x=\dfrac{3-m}{m-2}\left(m\ne2\right)\Rightarrow A\left(\dfrac{3-m}{m-2};0\right)\)
\(\left(d\right)\cap\left(Oy\right)\Leftrightarrow x=0\Leftrightarrow y=m-3\Rightarrow B\left(0;m-3\right)\)
\(OA=\sqrt{\left(\dfrac{3-m}{m-2}\right)^2}=\left|\dfrac{3-m}{m-2}\right|\)
\(OB=\sqrt{\left(m-3\right)^2}=\left|m-3\right|\)
\(S_{OAB}=\dfrac{1}{2}OA.OB=6\)
\(\Leftrightarrow\left|\dfrac{3-m}{m-2}\right|.\left|m-3\right|=12\)
\(\Leftrightarrow\left(m-3\right)^2=12\left|m-2\right|\)
\(\Leftrightarrow\left\{{}\begin{matrix}m\ge2\\\left(m-3\right)^2=12\left(m-2\right)\end{matrix}\right.\) \(\cup\left\{{}\begin{matrix}m< 2\\\left(m-3\right)^2=12\left(2-m\right)\end{matrix}\right.\)
\(TH1:\left\{{}\begin{matrix}m\ge2\\m^2-6m-33=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m\ge2\\m=3\pm\sqrt{42}\end{matrix}\right.\) \(\Leftrightarrow m=3+\sqrt{42}\)
\(TH2:\left\{{}\begin{matrix}m< 2\\m^2+6m-15=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m< 2\\m=-3\pm\sqrt{24}\end{matrix}\right.\) \(\Leftrightarrow m=-3-\sqrt{24}\)
Vậy \(m\in\left\{-3-\sqrt{24};3+\sqrt{42}\right\}\)
