Question

Fie z1=4a-i si z2=1-ai, a∈R. Sa se det a stiind ca z1/z2∈R

Answer 1

[tex]\frac{z_1}{z_2}=^{1+ai)}\frac{4a-1}{1-ai}=\frac{(4a-i)(1+ai)}{1+a^2}=\frac{4a+4a^2i-i-ai^2}{1+a^2}=\frac{5a+i(4a^2-1)}{1+a^2}=\\ \frac{5a}{1+a^2}+i\frac{4a^2-1}{1+a^2}.\\ \frac{z_1}{z_2}\in\mathbb{R}, daca \quad Im(z)=0, 4a^2-1=0, a^2=\frac{1}{4}, a= + sau - \frac{\sqrt{2}}{2}[/tex]