Gọi $M\left( {{x}_{0}};{{y}_{0}} \right),\left( {{x}_{0}}\ne 1 \right),{{y}_{0}}=\frac{2{{x}_{0}}+1}{{{x}_{0}}-1}$, Ta có $d\left( M,{{\Delta }_{1}} \right)=d\left( M,Ox \right)\Leftrightarrow \left| {{x}_{0}}-1 \right|=\left| {{y}_{0}} \right|$

$\Leftrightarrow \left| {{x}_{0}}-1 \right|=\left| \frac{2{{x}_{0}}+1}{{{x}_{0}}-1} \right|\Leftrightarrow {{\left( {{x}_{0}}-1 \right)}^{2}}=\left| 2{{x}_{0}}+1 \right|$

Với ${{x}_{0}}\ge \frac{-1}{2},$ ta có: $x_{0}^{2}-2{{x}_{0}}+1=2{{x}_{0}}+1\Leftrightarrow \left[ \begin{align}   & {{x}_{0}}=0 \\  & {{x}_{0}}=4 \\ \end{align} \right.$

Suy ra $M\left( 0;-1 \right),M\left( 4;3 \right)$

Với ${{x}_{0}}<\frac{-1}{2}$, ta có pt: $x_{0}^{2}-2{{x}_{0}}+1=-2{{x}_{0}}-1\Leftrightarrow x_{0}^{2}+2=0$ (vô nghiệm)