Question

Sa se determine m e R pentru care punctele A(m+1,2m) , B(3m+1,-1) C(m,2m-4) sunt coliniare.

Answer 1

$\displaystyle \mathtt{A(m+1;2m);~B(3m+1;-1);~C(m;2m-4)}\\ \\ \mathtt{Punctele~A,~B,~C~sunt~coliniare\Leftrightarrow \left|\begin{array}{ccc}\mathtt{m+1}&\mathtt{2m}&\mathtt1\\\mathtt{3m+1}&\mathtt{-1}&\mathtt1\\\mathtt m&\mathtt{2m-4}&\mathtt1\end{array}\right|=0\Leftrightarrow}$

$\displaystyle \mathtt{(m+1)\cdot(-1)\cdot1+1\cdot(3m+1)\cdot(2m-4)+2m\cdot1\cdot m-1\cdot(-1)\cdot m-}\\ \\ \mathtt{-2m\cdot(3m+1)\cdot1-(m+1)\cdot1\cdot(2m-4)=0}\\ \\ \mathtt{-m-1+6m^2-10m-4+2m^2+m-6m^2-2m-2m^2+2m+4=0}\\ \\ \mathtt{-10m-1=0}\\ \\ \mathtt{-10m=1}\\ \\ \mathtt{m=- \frac{1}{10} }$