Question Number 203417 by hardmath last updated on 18/Jan/24
$$\mathrm{If} \\ $$$$\mid\overline {−\mathrm{z}\:+\:\boldsymbol{\mathrm{i}}\mathrm{z}}\mid\:=\:\mathrm{2}\:\sqrt{\mathrm{13}}\:\:\:\mathrm{and}\:\:\:\mathrm{z}\:=\:\mathrm{1}\:+\:\left(\mathrm{x}\:+\:\mathrm{1}\right)\centerdot\boldsymbol{\mathrm{i}} \\ $$$$ \\ $$$$\mathrm{Find}:\:\:\:\mathrm{mun}\left(\mathrm{x}\right)\:=\:? \\ $$
Commented by hardmath last updated on 18/Jan/24
$$\boldsymbol{\mathrm{min}}\:\left(\mathrm{x}\right)\:=\:? \\ $$
Answered by Frix last updated on 19/Jan/24
$$\forall{r}\in\mathbb{C}:\:\mid{r}\mid=\mid\bar {{r}}\mid \\ $$$${r}=−{z}+\mathrm{i}{z}=−{x}−\mathrm{2}−{x}\mathrm{i} \\ $$$$\mid\bar {{r}}\mid=\sqrt{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}\right)}=\mathrm{2}\sqrt{\mathrm{13}}\:\Rightarrow\:{x}=−\mathrm{6}\vee{x}=\mathrm{4} \\ $$
Commented by hardmath last updated on 19/Jan/24
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{ser} \\ $$