Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers3 Marks
Question
Solve the system of equations $\text{Re}(\text{z}^2)=0,|\text{z}|=2.$
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Answer
$\text{Re}(\text{z}^2)=0,|\text{z}|=2$ Let $\text{z}=\text{x}+\text{iy}$ $\text{z}^2=0$ $\Rightarrow(\text{x}+\text{iy})^2=0$ $\Rightarrow\text{x}^2+2\text{ixy}-\text{y}^2=0$ $\Rightarrow\text{x}^2-\text{y}^2=0 \ ...(\text{i}),$ which is the real part of $(\text{x}+\text{iy})^2.$ $|\text{z}|=2$ $\Rightarrow\sqrt{\text{x}^2+\text{y}^2}=2$ $\Rightarrow\text{x}^2+\text{y}^2=4 \ ...(\text{ii})$ Adding (i) and (ii), we get $2\text{x}^2=4$ $\Rightarrow\text{x}^2=2$ $\Rightarrow\text{x}=\pm\sqrt{2},\text{y}=\pm\sqrt{2}$ $\text{x}+\text{iy}=\sqrt{2}+\text{i}\sqrt{2}$ $=\sqrt{2}-\text{i}\sqrt{2}$ $=\sqrt{2}+\text{i}\sqrt{2}$
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