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find-Z-if-arg-z-3-pi-and-arg-z-i-pi-4-




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

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