Find the conjugates of the following complex numbers: (1+i)(2+i) … - Vidyadip

Question

Find the conjugates of the following complex numbers:
$\frac{(1+\text{i})(2+\text{i})}{3+\text{i}}$

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Answer

Let $\text{Z}=\frac{(1+\text{i})(2+\text{i})}{3+\text{i}}$
$=\frac{2+\text{i}+\text{i}(2+\text{i})}{3+\text{i}}$
$=\frac{2+\text{i}+2\text{i}+\text{i}}{3+\text{i}}$
$=\frac{1+3\text{i}}{3+\text{i}}$
$=\frac{(1+3\text{i})}{(3+\text{i})}\times\frac{(3-\text{i})}{(3-\text{i})}$
$=\frac{3-\text{i}+3\text{i}(3-\text{i})}{3^2+1^2}$
$=\frac{3-\text{i}+9\text{i}+3}{9+1}$
$=\frac{6+8\text{i}}{10}$
$=\frac{2(3+4\text{i})}{10}$
$\Rightarrow\text{z}=\frac{3+4\text{i}}{5}$
Hence
$\bar{\text{z}}=\frac{3-4\text{i}}{5}$
$=\frac{3}{5}-\frac{4}{5}\text{i}$

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