The locus of a point on the Argand plane represented by the complex number $\mathrm{z},$ when $\mathrm{z}$ satisfies the condition

$\left|\frac{\mathrm{z}-1+\mathrm{i}}{\mathrm{z}+1-\mathrm{i}}\right|=\left|\mathrm{Re}\left(\frac{\mathrm{z}-1+\mathrm{i}}{\mathrm{z}+1-\mathrm{i}}\right)\right|$ is

$\left|\frac{\mathrm{z} - 1 + \mathrm{i}}{\mathrm{z} + 1 -\mathrm{i}}\right| = {\mathrm{R}}_{\mathrm{e}} \left(\frac{\mathrm{z}-1+\mathrm{i}}{\mathrm{z} + 1-\mathrm{i}}\right)$

$$\begin{gathered} \mathrm{substitute} z=x+iy \\ \Rightarrow \left|\frac{(\mathrm{x}-1) + \mathrm{i}(\mathrm{y} + 1)}{(\mathrm{z}+1) + \mathrm{i}(\mathrm{y}-1)}\right| = {\mathrm{R}}_{\mathrm{e}}\left|\frac{(\mathrm{x}-1) + \mathrm{i}(\mathrm{y}+1)}{(\mathrm{x}+1) + \mathrm{i}(\mathrm{y}-1)}\right| \end{gathered}$$

$\Rightarrow \frac{\sqrt{{(\mathrm{x}-1)}^2 + {(\mathrm{y} + 1)}^2}}{\sqrt{{(\mathrm{x}+1)}^2 + {(\mathrm{y} - 1)}^2}} = {\mathrm{R}}_{\mathrm{e}}\left(\frac{\left(\right(\mathrm{x}-1) + \mathrm{i}(\mathrm{y}+1\left)\right) \left(\right(\mathrm{x}+1) - \mathrm{i}(\mathrm{y}-1\left)\right)}{\left(\right(\mathrm{x}+1) + \mathrm{i}(\mathrm{y}-1\left)\right) \left(\right(\mathrm{x}+1) - \mathrm{i}(\mathrm{y}-1\left)\right)}\right)$

$\Rightarrow \frac{\sqrt{{(\mathrm{x}-1)}^2 + {(\mathrm{y} + 1)}^2}}{\sqrt{{(\mathrm{x}+1)}^2 + {(\mathrm{y} - 1)}^2}} = \frac{{\mathrm{x}}^2 + {\mathrm{y}}^2 -2}{{(\mathrm{x}+1)}^2 + {(\mathrm{y}-1)}^2}$

$\Rightarrow \sqrt{{(\mathrm{x}-1)}^2 + {(\mathrm{y} + 1)}^2} \sqrt{{(\mathrm{x}+1)}^2 + {(\mathrm{y} - 1)}^2} = {\mathrm{x}}^2 + {\mathrm{y}}^2 -2$

$\Rightarrow \left(\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + 2\right) - 2(\mathrm{x}-\mathrm{y})\right)\left(\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + 2\right) + 2(\mathrm{x}-\mathrm{y})\right) = {({\mathrm{x}}^2 + {\mathrm{y}}^2 -2)}^2$

$\Rightarrow {\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + 2\right)}^2-4{(\mathrm{x}-\mathrm{y})}^2+2(x-y)-{({\mathrm{x}}^2 + {\mathrm{y}}^2 -2)}^2=0$

$\Rightarrow 4\left({\mathrm{x}}^2 + {\mathrm{y}}^2+2\mathrm{xy}\right)=0\Rightarrow {\left(x+y\right)}^2=0\Rightarrow x+y=0$

is a Straight line