Question Number 80889 by M±th+et£s last updated on 07/Feb/20
$${find}\:{Z} \\ $$$${if}\:{arg}\left(\mathrm{z}−\mathrm{3}\right)=\pi \\ $$$${and}\:{arg}\left(\mathrm{z}+\mathrm{i}\right)=\frac{\pi}{\mathrm{4}} \\ $$
Commented by mr W last updated on 07/Feb/20
$${z}=\mathrm{1} \\ $$
Commented by msup trace by abdo last updated on 08/Feb/20
$$\Rightarrow{z}−\mathrm{3}={r}\:{e}^{{i}\pi} \:=−{r}\:\Rightarrow{z}=\mathrm{3}−{r}\:\:\left({r}>\mathrm{0}\right) \\ $$$${and}\:{z}+{i}=\rho\:{e}^{\frac{{i}\pi}{\mathrm{4}}} =\rho\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}+\frac{{i}}{.\sqrt{\mathrm{2}}}\right)\:\Rightarrow \\ $$$${z}=\frac{\rho}{\:\sqrt{\mathrm{2}}}\:+\left(\frac{\rho}{\:\sqrt{\mathrm{2}}}−\mathrm{1}\right){i}=\mathrm{3}−{r}\:\Rightarrow \\ $$$$\mathrm{3}−{r}=\frac{\rho}{\:\sqrt{\mathrm{2}}}\:{and}\:\frac{\rho}{\:\sqrt{\mathrm{2}}}−\mathrm{1}=\mathrm{0}\:\Rightarrow \\ $$$$\rho=\sqrt{\mathrm{2}}\:\Rightarrow\mathrm{3}−{r}=\mathrm{1}\:\Rightarrow{r}=\mathrm{2}\Rightarrow{z}=\mathrm{1} \\ $$
Commented by peter frank last updated on 08/Feb/20
$${thank}\:{you} \\ $$