解：$(2)$由题意得$x_{1}+m=y_{1}$，$\frac n{x_{1}}=y_{1}$，且$ x_{1}≠0 $
∴$x_{1}+m=\frac {n}{x_{1}}$，即$x_{1}^2+mx_{1}=n$
同理，得$x_{2}²+mx_{2}=n$
∴$x_{1}^2+mx_{1}=x_{2}²+mx_{2}$，
即$x_{1}^2-x_{2}^2+m(x_{1}-x_{2})=0$
∴$(x_{1}-x_{2})(x_{1}+x_{2})+m(x_{1}-x_{2})=0$
即$(x_{1}-x_{2}) · (x_{1}+x_{2}+m)=0$
通过图像易知$A$，$B$两点在$y$轴两侧
∴$x_{1}≠x_{2}$
∴$x_{1}+x_{2}+m=0$，即$x_{1}+x_{2}=-m$
∴$y_{1}+y_{2}=x_{1}+m+x_{2}+m=-m+2m=m$
又$m≠0$，∴$\frac {y_{1}+y_{2}}{x_{1}+x_{2}}=\frac {m}{-m}=-1$
