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Question Number 203417 by hardmath last updated on 18/Jan/24

$$\mathrm{If}$$$$\mid \overline{-\mathrm{z}+\mathbf{i}\mathrm{z}}\mid =2\sqrt{13}\mathrm{and}\mathrm{z}=1+(\mathrm{x}+1)\cdot \mathbf{i}$$$$$$$$\mathrm{Find}:\mathrm{mun}\left(\mathrm{x}\right)=?$$

Commented by hardmath last updated on 18/Jan/24

$$\mathbf{min}\left(\mathrm{x}\right)=?$$

Answered by Frix last updated on 19/Jan/24

$$\mathrm{\forall }r\in \mathbb{C}:\mid r\mid =\mid \overline{r}\mid$$$$r=-z+\mathrm{i}z=-x-2-x\mathrm{i}$$$$\mid \overline{r}\mid =\sqrt{2(x^2+2x+2)}=2\sqrt{13}\Rightarrow x=-6\vee x=4$$

Commented by hardmath last updated on 19/Jan/24

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{ser}$$

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