Question Number 122659 by mathocean1 last updated on 18/Nov/20
![Determinate the geometric aspect described by M(z) in complex plane such that: arg(z^− −3+i)≡(π/4)[2π]](Q122659.png)
$${Determinate}\:{the}\:{geometric} \\ $$$${aspect}\:{described}\:{by}\:{M}\left({z}\right)\:{in} \\ $$$${complex}\:{plane}\:{such}\:{that}: \\ $$$${arg}\left(\overset{−} {{z}}−\mathrm{3}+{i}\right)\equiv\frac{\pi}{\mathrm{4}}\left[\mathrm{2}\pi\right] \\ $$
Answered by MJS_new last updated on 18/Nov/20
![arg (x) ≡(π/4)[2π] ⇔ x=c+ci let z=a+bi ⇒ z^− =a−bi a−bi−3+i=c+ci ⇒ { ((a−c−3=0 ⇒ c=a−3)),((−b−c+1=0 ⇒ c=1−b)) :} a−3=1−b ⇒ b=4−a z=a+(4−a)i this is a straight line.](Q122684.png)
$$\mathrm{arg}\:\left({x}\right)\:\equiv\frac{\pi}{\mathrm{4}}\left[\mathrm{2}\pi\right]\:\Leftrightarrow\:{x}={c}+{c}\mathrm{i} \\ $$$$\mathrm{let}\:{z}={a}+{b}\mathrm{i}\:\Rightarrow\:\overset{−} {{z}}={a}−{b}\mathrm{i} \\ $$$${a}−{b}\mathrm{i}−\mathrm{3}+\mathrm{i}={c}+{c}\mathrm{i} \\ $$$$\Rightarrow\:\begin{cases}{{a}−{c}−\mathrm{3}=\mathrm{0}\:\Rightarrow\:{c}={a}−\mathrm{3}}\\{−{b}−{c}+\mathrm{1}=\mathrm{0}\:\Rightarrow\:{c}=\mathrm{1}−{b}}\end{cases} \\ $$$${a}−\mathrm{3}=\mathrm{1}−{b}\:\Rightarrow\:{b}=\mathrm{4}−{a} \\ $$$${z}={a}+\left(\mathrm{4}−{a}\right)\mathrm{i} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}. \\ $$