Tọa độ A là:
\(\left\{{}\begin{matrix}y=0\\\left(2m-1\right)x+m-6=0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}y=0\\x\left(2m-1\right)=-m+6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{-m+6}{2m-1}\\y=0\end{matrix}\right.\)
Vậy: \(A\left(\dfrac{-m+6}{2m-1};0\right)\)
Tọa độ B là:
\(\left\{{}\begin{matrix}x=0\\y=\left(2m-1\right)x+m-6\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x=0\\y=0\left(2m-1\right)+m-6=m-6\end{matrix}\right.\)
=>B(0;m-6)
\(O\left(0;0\right);A\left(\dfrac{-m+6}{2m-1};0\right);B\left(0;m-6\right)\)
\(OA=\sqrt{\left(\dfrac{-m+6}{2m-1}-0\right)^2+\left(0-0\right)^2}=\left|\dfrac{m-6}{2m-1}\right|\)
\(OB=\sqrt{\left(0-0\right)^2+\left(m-6-0\right)^2}\)
\(=\sqrt{\left(m-6\right)^2}=\left|m-6\right|\)
OA=2OB
=>\(\dfrac{\left|m-6\right|}{\left|2m-1\right|}=2\left|m-6\right|\)
=>\(\dfrac{\left|m-6\right|}{\left|2m-1\right|}-2\left|m-6\right|=0\)
=>\(\left|m-6\right|\left(\dfrac{1}{\left|2m-1\right|}-2\right)=0\)
=>\(\left[{}\begin{matrix}m-6=0\\\left|2m-1\right|=\dfrac{1}{2}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m-6=0\\2m-1=\dfrac{1}{2}\\2m-1=-\dfrac{1}{2}\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}m=6\left(nhận\right)\\m=\dfrac{3}{4}\left(nhận\right)\\m=\dfrac{1}{4}\left(nhận\right)\end{matrix}\right.\)
