Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 623

\[\left\{ \begin{matrix} \left| \frac{z - 4}{z - 8} \right| - 1 = 0 \\ \left| \frac{z - 12}{z - 8i} \right| = \frac{5}{3}\text{\ \ \ \ \ \ } \\ \end{matrix} \right.\ \]

\[1)\ \left| \frac{z - 4}{z - 8} \right| - 1 = 0\]

\[\left| \frac{z - 4}{z - 8} \right| = 1\]

\[|z - 4| = |z - 8|\]

\[(x - 4)^{2} + y^{2} = (x - 8)^{2} + y^{2}\]

\[x^{2} - 8x + 16 = x^{2} - 16x + 64\]

\[8x = 48\]

\[x = 6.\]

\[2)\ \left| \frac{z - 12}{z - 8i} \right| = \frac{5}{3}\ \]

\[3|z - 12| = 5|z - 8i|\]

\[9\left( (x - 12)^{2} + y^{2} \right) =\]

\[= 25\left( x^{2} + (y - 8)^{2} \right)\]

\[9\left( (6 - 12)^{2} + y^{2} \right) =\]

\[= 25\left( 6^{2} + y^{2} - 16y + 64 \right)\]

\[9\left( 36 + y^{2} \right) = 25\left( y^{2} - 16y + 100 \right)\]

\[324 + 9y^{2} = 25y^{2} - 400y + 2500\]

\[16y^{2} - 400y + 2176 = 0\]

\[y^{2} - 25y + 136 = 0\]

\[D = 625 - 544 = 81\]

\[y_{1} = \frac{25 - 9}{2} = 8;\ \]

\[y_{2} = \frac{25 + 9}{2} = 17.\]

\[Ответ:\ \ z_{1} = 6 + 8i;\ \]

\[\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }z_{2} = 6 + 17i.\]