Question

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$|z + w| = \sqrt{3} \\ |z| = |w| = 1 \\ |z - w | =$
aflati |z-w|

Answer 1

[tex]\text{Folosim urmatoare identitate:}\\ \text{Daca }z\in \mathbb{C\backslash R},\text{atunci: }z\cdot \overline{z}=|z|^2\\ \text{Avem ca:}\\ |z+w|=\sqrt 3|()^2\\ (z+w)(\overline{z}+\overline{w})=3\\ z\cdot \overline{z}+z\cdot \overline{w}+w\cdot \overline{z}+w\cdot \overline{w}=3\\ |z|^2+z\cdot \overline{w}+w\cdot \overline{z}+|w|^2=3\\ z\cdot \overline{w}+w\cdot \overline{z}+2=3\\ z\cdot \overline{w}+w\cdot \overline{z}=1(\text{relatia asta ne va ajuta mai tarziu})\\ \text{Notez }|z-w|=l\\ l^2=|z-w|^2\\ [/tex]
[tex]l^2=(z-w)(\overline{z}-\overline{w})\\ l^2=z\cdot \overline{z}-z\cdot \overline{w}-w\cdot \overline{z}+w\cdot \overline{w}\\ l^2=|z|^2+|w|^2-(z\cdot \overline{w}+w\cdot \overline{z})\\ l^2=1+1-1\\ l^2=1\Rightarrow l=1\\ \text{Deci }|z-w|=1[/tex]