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Câu hỏi: Cho số phức z thỏa mãn $\dfrac{\overline{z}}{4-3i}+\left( 2-3i \right)=5-2i$. Tìm mô đun của số phức z
A. $\left| z \right|=5\sqrt{10}$
B. $\left| z \right|=10\sqrt{2}$
C. $\left| z \right|=\sqrt{10}$
D. $\left| z \right|=2\sqrt{10}$
A. $\left| z \right|=5\sqrt{10}$
B. $\left| z \right|=10\sqrt{2}$
C. $\left| z \right|=\sqrt{10}$
D. $\left| z \right|=2\sqrt{10}$
Cách 1:
$\dfrac{\overline{z}}{4-3i}+\left( 2-3i \right)=5-2i\Leftrightarrow \dfrac{\overline{z}}{4-3i=3+i}\Leftrightarrow \overline{z}=\left( 4-3i \right)\left( 3+i \right)\Leftrightarrow \overline{z}=15-5i$
$\Rightarrow z=15+5i\Rightarrow \left| z \right|=\sqrt{{{15}^{2}}+{{5}^{2}}}=5\sqrt{10}$
Cách 2:
$\dfrac{\overline{z}}{4-3i}+\left( 2-3i \right)=5-2i\Leftrightarrow \dfrac{\overline{z}}{4-3i}=3+i\Leftrightarrow \overline{z}=\left( 4-3i \right)\left( 3+i \right)\left( 3+i \right)\Rightarrow \left| \overline{z} \right|=\left| \left( 4-3i \right)\left( 3+i \right) \right|$
$\Rightarrow \left| z \right|=\left| 4-3i \right|.\left| 3+i \right|=\sqrt{{{4}^{2}}+{{\left( -3 \right)}^{2}}}.\sqrt{{{3}^{2}}+{{1}^{2}}}=5\sqrt{10}$
$\dfrac{\overline{z}}{4-3i}+\left( 2-3i \right)=5-2i\Leftrightarrow \dfrac{\overline{z}}{4-3i=3+i}\Leftrightarrow \overline{z}=\left( 4-3i \right)\left( 3+i \right)\Leftrightarrow \overline{z}=15-5i$
$\Rightarrow z=15+5i\Rightarrow \left| z \right|=\sqrt{{{15}^{2}}+{{5}^{2}}}=5\sqrt{10}$
Cách 2:
$\dfrac{\overline{z}}{4-3i}+\left( 2-3i \right)=5-2i\Leftrightarrow \dfrac{\overline{z}}{4-3i}=3+i\Leftrightarrow \overline{z}=\left( 4-3i \right)\left( 3+i \right)\left( 3+i \right)\Rightarrow \left| \overline{z} \right|=\left| \left( 4-3i \right)\left( 3+i \right) \right|$
$\Rightarrow \left| z \right|=\left| 4-3i \right|.\left| 3+i \right|=\sqrt{{{4}^{2}}+{{\left( -3 \right)}^{2}}}.\sqrt{{{3}^{2}}+{{1}^{2}}}=5\sqrt{10}$
Đáp án A.