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МатематикаРешебник по алгебре и начала математического анализа 11 класс Колягин Задание 683
Задание 683
\[1)\ \overline{z_{1}z_{2}} = \overline{z_{1}} \bullet \overline{z_{2}}\]
\[\overline{z_{1}z_{2}} = \overline{\left( x_{1} + iy_{1} \right)\left( x_{2} + iy_{2} \right)} =\]
\[= \overline{x_{1}x_{2} + x_{1}y_{2}i + x_{2}y_{1}i - y_{1}y_{2}} =\]
\[= \overline{\left( x_{1}x_{2} - y_{1}y_{2} \right) + \left( x_{1}y_{2} + x_{2}y_{1} \right)i} =\]
\[= \left( x_{1}x_{2} - y_{1}y_{2} \right) - \left( x_{1}y_{2} + x_{2}y_{1} \right)i;\]
\[\overline{z_{1}} \bullet \overline{z_{2}} = \overline{x_{1} + iy_{1}} \bullet \overline{x_{2} + iy_{2}} =\]
\[= \left( x_{1} - iy_{1} \right)\left( x_{2} - iy_{2} \right) =\]
\[= x_{1}x_{2} - x_{1}y_{2}i - x_{2}y_{1}i - y_{1}y_{2} =\]
\[= \left( x_{1}x_{2} - y_{1}y_{2} \right) - \left( x_{1}y_{2} + x_{2}y_{1} \right)i.\]
\[Что\ и\ требовалось\ доказать.\]
\[2)\ \overline{\left( \frac{z_{1}}{z_{2}} \right)} = \frac{\overline{z_{1}}}{\overline{z_{2}}};\text{\ \ \ }z_{2} \neq 0:\]
\[\ \overline{\left( \frac{z_{1}}{z_{2}} \right)} = \overline{\frac{x_{1} + iy_{1}}{x_{2} + iy_{2}}} =\]
\[= \overline{\frac{\left( x_{1} + iy_{1} \right)\left( x_{2} - iy_{2} \right)}{\left( x_{2} + iy_{2} \right)\left( x_{2} - iy_{2} \right)}} =\]
\[= \overline{\frac{x_{1}x_{2} - x_{1}y_{2}i + x_{2}y_{1}i + y_{1}y_{2}}{x_{2}^{2} + y_{2}^{2}}} =\]
\[= \overline{\frac{\left( x_{1}x_{2} + y_{1}y_{2} \right) + \left( x_{2}y_{1} - x_{1}y_{2} \right)i}{x_{2}^{2} + y_{2}^{2}}} =\]
\[= \frac{\left( x_{1}x_{2} + y_{1}y_{2} \right) - \left( x_{2}y_{1} - x_{1}y_{2} \right)i}{x_{2}^{2} + y_{2}^{2}};\]
\[\frac{\overline{z_{1}}}{\overline{z_{2}}} = \frac{\overline{x_{1} + iy_{1}}}{\overline{x_{2} + iy_{2}}} = \frac{x_{1} - iy_{1}}{x_{2} - iy_{2}} =\]
\[= \frac{\left( x_{1} - iy_{1} \right)\left( x_{2} + iy_{2} \right)}{\left( x_{2} - iy_{2} \right)\left( x_{2} + iy_{2} \right)} =\]
\[= \frac{x_{1}x_{2} + x_{1}y_{2}i - x_{2}y_{1}i + y_{1}y_{2}}{x_{2}^{2} + y_{2}^{2}} =\]
\[= \frac{\left( x_{1}x_{2} + y_{1}y_{2} \right) - \left( x_{2}y_{1} - x_{1}y_{2} \right)i}{x_{2}^{2} + y_{2}^{2}}.\]
\[Что\ и\ требовалось\ доказать.\]
