The equation $\left|\mathrm{z} - \mathrm{i}\right| = \left|\mathrm{z} -1\right|, \mathrm{i} = \sqrt{-1},$represents 

$$\begin{gathered} \left|\mathrm{z}-\mathrm{i}\right| = \left|\mathrm{z} - 1\right| , \mathrm{z} = \mathrm{x} + \mathrm{iy} \\ \left|\mathrm{x} + \mathrm{iy} - i\right| = \left|\mathrm{x} + \mathrm{iy} - 1\right| \\ \left|\mathrm{x} + \mathrm{i}\left(\mathrm{y}-1\right)\right| = \left|\left(\mathrm{x} - 1\right) + \mathrm{iy}\right| \\ \sqrt{{\mathrm{x}}^2 + {\left(\mathrm{y}-1\right)}^2} = \sqrt{{\left(\mathrm{x}-1\right)}^2 + {\mathrm{y}}^2} \\ {\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{y} + 1 = {\mathrm{x}}^2 - 2\mathrm{x} + 1 + {\mathrm{y}}^2 \\ 2\mathrm{x} - 2\mathrm{y} = 0 \\ \mathrm{x} - \mathrm{y} = 0 \end{gathered}$$

is the line passing through (0, 0 ) with slope1