Let z be a complex number satisfying arg ((z-1)/(z+1))=(π)/(4). Let m and M be the least and the greatest values of |z-2| as z varies over all such…

Put $z=x+iy$. $\arg\frac{z-1}{z+1}=\arg(z-1)-\arg(z+1)=\frac\pi4$ means the segment from $-1$ to $1$ subtends an angle $\frac\pi4$ at $z$ (anticlockwise sense), so $z$ lies on the upper arc of the circle through $\pm1$ whose inscribed angle is $\frac\pi4$. The central angle is $\frac\pi2$, so the centre is $C(0,1)$ and the radius is $\sqrt2$; the locus is the part with $y>0$ (check: $z=i(1+\sqrt2)$ gives $\arg=\pi-2\tan^{-1}(1+\sqrt2)=\frac\pi4$). The point $P(2,0)$ is at distance $\sqrt5$ from $C$. The nearest and farthest points on the circle lie on line $PC$ at $(\sqrt2/\sqrt5)$ along it, with $y=1\mp\sqrt{2/5}>0$, so both belong to the arc. Hence $m=\sqrt5-\sqrt2$, $M=\sqrt5+\sqrt2$ and $mM=5-2=3$.