Tọa độ giao điểm của (d1) và (d2) là:
\(\left\{{}\begin{matrix}mx+2y=m+1\\2x+my=2m-1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}2mx+4y=2m+2\\2mx+m^2y=2m^2-m\end{matrix}\right.0\Leftrightarrow\left\{{}\begin{matrix}2mx+m^2y-2mx-4y=2m^2-m-2m-2\\mx+2y=m+1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}y\left(m^2-4\right)=2m^2-3m-2\\mx+2y=m+1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}y\left(m-2\right)\left(m+2\right)=\left(m-2\right)\left(2m+1\right)\\mx+2y=m+1\end{matrix}\right.\)(1)
TH1: m=2
Hệ phương trình (1) sẽ trở thành:
\(\left\{{}\begin{matrix}y\left(2-2\right)\left(2+2\right)=\left(2-2\right)\left(2\cdot2+1\right)\\2x+2y=2+1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}0y=0\\2x+2y=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y\in R\\2x+2y=3\end{matrix}\right.\)
Vậy: Khi m=2 thì (d1) và (d2) trùng nhau
TH2: m=-2
Hệ phương trình (1) sẽ trở thành:
\(\left\{{}\begin{matrix}y\cdot\left(-2-2\right)\left(-2+2\right)=\left(-2-2\right)\left(-2\cdot2+1\right)\\-2x+2y=-2+1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}0y=\left(-4\right)\cdot\left(-3\right)=12\\-2x+2y=-1\end{matrix}\right.\Leftrightarrow\left(x,y\right)\in\varnothing\)
Vậy: Khi m=-2 thì (d1)//(d2)
TH3: \(m\notin\left\{2;-2\right\}\)
hệ phương trình (1) sẽ trở thành:
\(\left\{{}\begin{matrix}y=\dfrac{\left(m-2\right)\left(2m+1\right)}{\left(m-2\right)\left(m+2\right)}=\dfrac{2m+1}{m+2}\\mx+2y=m+1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}y=\dfrac{2m+1}{m+2}\\mx=m+1-\dfrac{4m+2}{m+2}=\dfrac{\left(m+1\right)\left(m+2\right)-4m-2}{m+2}\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}y=\dfrac{2m+1}{m+2}\\x=\dfrac{m^2+3m+2-4m-2}{m\left(m+2\right)}=\dfrac{m^2-m}{m\left(m+2\right)}=\dfrac{m-1}{m+2}\end{matrix}\right.\)
vậy: Khi \(m\notin\left\{2;-2\right\}\) thì (d1) cắt (d2) tại \(A\left(\dfrac{m-1}{m+2};\dfrac{2m+1}{m+2}\right)\)
