[tex] \it |z|=\Big|\dfrac{1}{z}\Big| \Rightarrow |z|= \dfrac{1}{|z|} \Rightarrow |z|^2=1 \Rightarrow |z| =1\\ \\ Fie\ z=a+bi \Rightarrow |z|=\sqrt{a^2+b^2}\\ \\ |z-1|=|a+bi-1| = |(a-1)+bi| = \sqrt{(a-1)^2+b^2}\\ \\ |z| =|z-1| \Rightarrow \sqrt{a^2+b^2} =\sqrt{(a-1)^2+b^2} \Rightarrow a^2+b^2=(a-1)^2+b^2\Rightarrow \\ \\ a^2=(a-1)^2 \Rightarrow a^2=a^2-2a+1\Rightarrow 2a=1\Rightarrow a=\dfrac{1}{2}[/tex]
[tex] \it |z| =1\Rightarrow\sqrt{a^2+b^2}=1 \Rightarrow a^2+b^2=1 \Rightarrow b^2=1-a^2 \Rightarrow b^2=1-\dfrac{1}{4}\Rightarrow \\ \\ \Rightarrow b^2=\dfrac{3}{4} \Rightarrow b=\pm\dfrac{\sqrt3}{2}\\ \\ Deci,\ \ z= \dfrac{1}{2} \pm\dfrac{\sqrt3}{2}i [/tex]
