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Câu hỏi: Cho số phức $z$ thỏa mãn $z\left( 1+3i \right)=1-4i+3z.$ Tính $\left| z \right|.$
A. $\left| z \right|=\dfrac{17}{13}.$
B. $\left| z \right|=\sqrt{\dfrac{17}{13}}.$
C. $\left| z \right|=\dfrac{13}{\sqrt{17}}.$
D. $\left| z \right|=\dfrac{\sqrt{13}}{17}.$
Ta có $z\left( 1+3i \right)=1-4i+3z\Leftrightarrow z\left( -2+3i \right)=1-4i\Leftrightarrow z=\dfrac{1-4i}{-2+3i}=-\dfrac{14}{13}+\dfrac{5}{13}i$
$\Rightarrow \left| z \right|=\left| -\dfrac{14}{13}+\dfrac{5}{13}i \right|=\sqrt{{{\left( -\dfrac{14}{13} \right)}^{2}}+{{\left( \dfrac{5}{13} \right)}^{2}}}=\sqrt{\dfrac{17}{13}}.$
A. $\left| z \right|=\dfrac{17}{13}.$
B. $\left| z \right|=\sqrt{\dfrac{17}{13}}.$
C. $\left| z \right|=\dfrac{13}{\sqrt{17}}.$
D. $\left| z \right|=\dfrac{\sqrt{13}}{17}.$
Ta có $z\left( 1+3i \right)=1-4i+3z\Leftrightarrow z\left( -2+3i \right)=1-4i\Leftrightarrow z=\dfrac{1-4i}{-2+3i}=-\dfrac{14}{13}+\dfrac{5}{13}i$
$\Rightarrow \left| z \right|=\left| -\dfrac{14}{13}+\dfrac{5}{13}i \right|=\sqrt{{{\left( -\dfrac{14}{13} \right)}^{2}}+{{\left( \dfrac{5}{13} \right)}^{2}}}=\sqrt{\dfrac{17}{13}}.$
Đáp án B.