The greatest positive argument of complex number satisfying |z - 4| = Re(z) is

$\begin{array}{l}|\mathrm{z}-4|=\mathrm{Re}\left(\mathrm{z}\right)\\ \Rightarrow \sqrt{(\mathrm{x}-4{)}^2+{\mathrm{y}}^2}=\mathrm{x}\end{array}$

$\begin{array}{l}\mathrm{or} {\mathrm{x}}^2-8\mathrm{x}+16+{\mathrm{y}}^2={\mathrm{x}}^2\\ \mathrm{or} {\mathrm{y}}^2=8(\mathrm{x}-2)\end{array}$

The given relation represents the parabola with focus (4, 0) and the imaginary axis as the directirx. The two tangents from directrix are at right angle. Hence, greatest positive argument of z is $\mathrm{\pi }/4$.