Let \[z\text{ }=\text{ }x\text{ }+\text{ }iy\]
arg \[\left( \left( z-1 \right)/\left( z+1 \right) \right)\text{ }=\text{ }\pi /4\]
\[\Rightarrow ~arg\text{ }\left( z\text{ }\text{ }1 \right)\text{ }\text{ }-arg\text{ }\left( z\text{ }+\text{ }1 \right)\text{ }=\text{ }\pi /4\]
\[\Rightarrow ~arg\text{ }\left( x\text{ }+\text{ }iy\text{ }\text{ }1 \right)\text{ }\text{ }-arg\text{ }\left( x\text{ }+\text{ }iy\text{ }+\text{ }1 \right)\text{ }=\text{ }\pi /4\]
\[\Rightarrow ~arg\text{ }\left( x+\text{ }\text{ }1\text{ }+\text{ }iy \right)\text{ }\text{ }-arg\text{ }\left( x\text{ }+\text{ }1\text{ }+\text{ }iy \right)\text{ }=\text{ }\pi /4\]
\[\Rightarrow ~{{x}^{2}}~+\text{ }{{y}^{2}}~\text{ }-1\text{ }=\text{ }2y\]
\[\Rightarrow ~{{x}^{2}}~+\text{ }{{y}^{2}}~\text{ }-2y\text{ }\text{ }-1\text{ }=\text{ }0\]
The equation obtained represents the equation of a circle.