Question

Se considera numerele complexe z si w astfel incat
|z| = |w| = radical din 1006
| z + w | = radical din 2013
sa se calculeze | z - w |

Answer 1

$|z| = |w| = \sqrt{1006} \\ \\ \sqrt{z\cdot \overline{z}} =\sqrt{w\cdot \overline{w}} =\sqrt{1006} \\z\cdot \overline{z}=w\cdot \overline{w} = 1006\\ \\ \\|z+w| = \sqrt{2013} \\\\ \sqrt{(z+w)\cdot (\overline{z+w})} = \sqrt{2013}\\ (z+w)\cdot (\overline{z}+\overline{w}) = 2013 \\ z\cdot \overline{z}+z\cdot \overline{w}+w\cdot \overline{z}+w\overline{w} = 2013 \\ 1006+z\cdot \overline{w}+w\cdot \overline{z}+1006 = 2013\\ \\z\cdot \overline{w}+w\cdot \overline{z} = 1$

$|z-w| =\sqrt{(z-w)\cdot(\overline{z-w})} = \sqrt{(z-w)\cdot(\overline{z}-\overline{w})} = \\ \\ =\sqrt{z\cdot \overline{z}-(z\cdot \overline{w}+w\cdot \overline{z})+w\cdot \overline{w}} = \sqrt{1006-1+1006} = \\ \\ =\sqrt{2011}$