Question

Considere o quociente:
$\frac{11 - 10i}{2 - 3i}$
a) calcule
$| \frac{11 - 10i}{2 - 3i} |$
b) calcule
$\frac{ |11 - 10i| }{ |2 - 3i| }$
​

Answer 1

Resposta:

Explicação passo-a-passo:

a) temos que

$\frac{11-10i}{2-3i}=\frac{(11-10i)(2-3i)}{(2-3i)(2+3i)}=\frac{22-33i-20i+30i^{2}}{2^{2}+3^{2}}=\frac{22-53i-30}{4+9}=\frac{-8-53i}{13}$. Logo:

$|-\frac{8}{13}-\frac{53}{13}|=\sqrt{(-\frac{8}{13})^{2}+(-\frac{53}{13})^{2}}=\sqrt{\frac{64}{169}+\frac{2809}{169}}=\sqrt{\frac{2873}{169}}=\sqrt{221}$

b)

$|11-10i|=\sqrt{11^{2}+10^{2}}=\sqrt{121+100}=\sqrt{221}$

$|2-3i|=\sqrt{2^{2}+3^{2}}=\sqrt{4+9}=\sqrt{13}$

Assim:

$\frac{|11-10i|}{|2-3i|}=\frac{\sqrt{221}}{\sqrt{13}}=\sqrt{\frac{221}{13}}=\sqrt{17}$