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Complex Numbers — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers4 Marks
Question
If $\text{z}_1=2-\text{i}, \ \text{z}_2=-2+\text{i},$ Find $\text{Re}\Big(\frac{\text{z}_1\text{z}_2}{\text{z}_1}\Big)$
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Answer

$\frac{\text{z}_1\text{z}_2}{\text{z}_1}=\frac{\text{z}_1\text{z}_2}{\text{z}_1}\times\frac{\text{z}_1}{\text{z}_1}$ (rationalising the denominator) $=\frac{(\text{z}_1)^2\text{z}_2}{\text{z}_1\text{z}_1}$ $=\frac{(2-\text{i})^2(-2-\text{i})}{|\text{z}_1|^2} \ \big(\therefore \ \text{z}\bar{\text{z}=|\text{z}|^2}\big)$ $=\frac{(2^2+\text{i}^2-2\times2\times\text{i})(-2+\text{i})}{|2-\text{i}|^2}$ $=\frac{(4-1-4\text{i}^2)(-2+\text{i})}{2^2+(-1)^2}$ $=\frac{(3-4\text{i})(-2+\text{i})}{4+\text{i}}$ $=3(-2+\text{i})-4\text{i}(-2+\text{i})$ $=\frac{-6+3\text{i}+8\text{i}+4}{5}$ $=\frac{-2+11\text{i}}{5}$ $\therefore \ \text{Re}\Big(\frac{\text{z}_1\text{z}_2}{\text{z}_1}\Big)=\text{Re}\Big(\frac{-2}{5}+\frac{11}{5}\text{i}\Big)$ $=\frac{-2}{5}$

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