If $\;z_{1}\;$ and $\;z_{2}\;$ both satisfy $\;z+\overline{z} =2|z-1|\;arg(z_{1}-z_{2}) = \large\frac{\pi}{4}\;,$ then find $\;Im(z_{1}+z_{2})\;.$

Question

$(a)\;2\qquad(b)\;\pm6\qquad(c)\;\pm3\qquad(d)\;4$