Finding reduced mass equation

Question

I want to find reduced mass using newtonian mechanics but I fail. Can someone help me to find my mistake?

My work :

I'm not sure why the final equation has a negative in left side but right side is positive.

Answer 1

$\def \b {\mathbf}$ you have two equations

$$ \b{\ddot{r}}_1=\frac{1}{m_1}\b F_{12}$$ $$ \b{\ddot{r}}_2=\frac{1}{m_2}\left(-\b F_{12}\right)$$

obtain equation (1) minus equation (2) $$\b{\ddot{r}}_1-\b{\ddot{r}}_2=\b{\ddot{r}}= \left(\frac{1}{m_1}+\frac{1}{m_2}\right)\,\b F_{12}\quad\Rightarrow\\ \mu\,\b{\ddot{r}}=-\frac{G\,m_1\,m_2}{\b r^2}\,\b{\hat{r}}\\ \mu=\frac{m_1\,m_2}{m_1+m_2}$$