Question

determine z\1+i+z\1-i=5,z=a+bi,a,b,E r calcule a,b,c

Answer 1

Queremos encontrar o valor da variável $z=a+bi$, onde $a,b \in \mathbb{R}$, e $z \in \mathbb{C}$.

$\frac{z}{1+i}+\frac{z}{1-i}=5 \ \ \rightarrow \times (1+i)(1-i)\\ \\ \left( \frac{z}{1+i}+\frac{z}{1-i}\right )(1+i)(1-i)=5(1+i)(1-i)\\ \\ \frac{z(1+i)(1-i)}{1+i}+\frac{z(1+i)(1-i)}{1-i}=5(1+i)(1-i)\\ \\ z(1-i)+z(1+i)=5(1+i)(1-i)\\ \\ z\left[ (1+i)+(1-i)\right ]=5(1-i+i-i^{2})\\ \\z \cdot 2 = 5(1-(-1))\\ \\2z=5 \cdot 2\\ \\z=\frac{5 \cdot \not 2}{\not 2} \Rightarrow z=5$.

Colocando $z$ na forma $a+bi$, temos

$z=\underbrace{5}_{\text{a}}+\underbrace{0}_{\text{b}}i$

Logo, $a=5 \text{ e }b=0$.