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\(\left\{{}\begin{matrix}x-my=2-4m\\m^2x+my=3m^2+m\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x-my=2-4m\\\left(m^2+1\right)x=3m^2-3m+2\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x=\frac{3m^2-3m+2}{m^2+1}=3-\frac{3m+1}{m^2+1}\\y=\frac{4m^2+m+1}{m^2+1}=4-\frac{3-m}{m^2+1}\end{matrix}\right.\)
\(L=\left(3-\frac{3m+1}{m^2+1}\right)^2+\left(4-\frac{3-m}{m^2+1}\right)^2-6+\frac{6m+2}{m^2+1}\)
\(=19-\frac{4m+6}{m^2+1}\)
\(L_{max}\) khi \(k=\frac{4m+6}{m^2+1}\) đạt min
\(k=\frac{4m+6}{m^2+1}=km^2-4m+k-6=0\)
\(\Delta'=4-k\left(k-6\right)\ge0\)
\(\Leftrightarrow-k^2+6k+4\ge0\Rightarrow3-\sqrt{13}\le k\le3+\sqrt{13}\)
\(\Rightarrow L\le19-\left(3-\sqrt{13}\right)=16+\sqrt{13}\)
\(\hept{\begin{cases}mx+y=m^2+m+1\\-x+my=m^2\end{cases}}\Leftrightarrow\hept{\begin{cases}m\left(my-m^2\right)+y-m^2-m-1=0\\x=my-m^2\end{cases}}\)
\(\Leftrightarrow\)\(\hept{\begin{cases}\left(m^2y-m^2\right)+\left(y-1\right)-\left(m^3+m\right)=0\\x=my-m^2\end{cases}}\Leftrightarrow\hept{\begin{cases}\left(m^2+1\right)\left(y-m-1\right)=0\\x=my-m^2\end{cases}}\)
\(\Leftrightarrow\)\(\hept{\begin{cases}y=m+1\\x=m\left(m+1\right)-m^2\end{cases}}\Leftrightarrow\hept{\begin{cases}x=m\\y=m+1\end{cases}}\)
\(\Rightarrow\)\(x^2+y^2=2m^2+2m+1=2\left(m+\frac{1}{2}\right)^2+\frac{1}{2}\ge\frac{1}{2}\)
Dấu "=" xảy ra khi \(m=\frac{-1}{2}\) hay hệ có nghiệm \(\left(x;y\right)=\left(\frac{-1}{2};\frac{1}{2}\right)\)
