if z=x+iy is a complex number such that |z|=Re(iz)+1, find the locus of z

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{z}=\mathrm{x}+\mathrm{iy} \\ \Rightarrow \left|\mathrm{z}\right|=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2} \\ \mathrm{iz}=\mathrm{ix}+{\mathrm{i}}^2\mathrm{y}=-\mathrm{y}+\mathrm{ix} \left[\mathrm{As} {\mathrm{i}}^2=-1\right] \\ \Rightarrow \mathrm{R}\left(\mathrm{iz}\right)=-\mathrm{y} \\ \mathrm{Given} \mathrm{that} \\ \left|\mathrm{z}\right|=\mathrm{Re}\left(\mathrm{iz}\right)+1 \\ \Rightarrow \sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}=-\mathrm{y}+1 \\ \mathrm{Squarring} \mathrm{both} \mathrm{sides} \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{y}}^2+1-2\mathrm{y} \\ \Rightarrow {\mathbf{x}}^{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{2}\mathbf{y} \\ \mathrm{which} \mathrm{is} \mathrm{a} \mathrm{locus} \mathrm{of} \mathrm{parabola}. \end{gathered}$$

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