Question Number 123886 by Ar Brandon last updated on 29/Nov/20
$$\mathrm{Let}\:\overset{−} {{l}z}+{l}\overset{−} {{z}}+{m}=\mathrm{0}\:\mathrm{be}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{in}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{plane} \\ $$$$\mathrm{and}\:{P}\left({z}_{\mathrm{0}} \right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{the}\:\mathrm{plane}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{passing}\:\mathrm{through}\:{P}\left({z}_{\mathrm{0}} \right)\:\mathrm{and}\:\mathrm{perpendicular} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{given}\:\mathrm{line}\:\mathrm{is}\:\_\_\_ \\ $$
Answered by Olaf last updated on 29/Nov/20
![Equation of the straight line : l^_ z+lz^_ +m = 0 Equation of any line passing through the given line : −lz+l^_ z^_ +q = 0 (1) For a perpendicular line passing through P(z_0 ) : −lz_0 +l^_ z_0 ^_ +q = 0 (2) (2)−(1) : l(z−z_0 )−l^_ (z^− −z_0 ^_ ) = 0 ⇒ Im[l(z−z_0 )] = 0 ⇒ l(z−z_0 )∈R](Q123892.png)
$$\mathrm{Equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{straight}\:\mathrm{line}\:: \\ $$$$\overset{\_} {{l}z}+{l}\overset{\_} {{z}}+{m}\:=\:\mathrm{0} \\ $$$$\mathrm{Equation}\:\mathrm{of}\:\mathrm{any}\:\mathrm{line}\:\mathrm{passing}\:\mathrm{through} \\ $$$$\mathrm{the}\:\mathrm{given}\:\mathrm{line}\:: \\ $$$$−{lz}+\overset{\_} {{l}}\overset{\_} {{z}}+{q}\:=\:\mathrm{0}\:\left(\mathrm{1}\right) \\ $$$$\mathrm{For}\:\mathrm{a}\:\mathrm{perpendicular}\:\mathrm{line}\:\mathrm{passing} \\ $$$$\mathrm{through}\:\mathrm{P}\left({z}_{\mathrm{0}} \right)\:: \\ $$$$−{lz}_{\mathrm{0}} +\overset{\_} {{l}}\overset{\_} {{z}}_{\mathrm{0}} +{q}\:=\:\mathrm{0}\:\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{2}\right)−\left(\mathrm{1}\right)\::\:{l}\left({z}−{z}_{\mathrm{0}} \right)−\overset{\_} {{l}}\left(\overset{−} {{z}}−\overset{\_} {{z}}_{\mathrm{0}} \right)\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{Im}\left[{l}\left({z}−{z}_{\mathrm{0}} \right)\right]\:=\:\mathrm{0} \\ $$$$\Rightarrow\:{l}\left({z}−{z}_{\mathrm{0}} \right)\in\mathbb{R} \\ $$