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Question Number 123886 by Ar Brandon last updated on 29/Nov/20

$$\mathrm{Let}\stackrel{-}{lz}+l\stackrel{-}{z}+m=0\mathrm{be}\mathrm{a}\mathrm{straight}\mathrm{line}\mathrm{in}\mathrm{the}\mathrm{complex}\mathrm{plane}$$$$\mathrm{and}P\left(z_0\right)\mathrm{be}\mathrm{a}\mathrm{point}\mathrm{in}\mathrm{the}\mathrm{plane}.\mathrm{Then}\mathrm{the}\mathrm{equation}$$$$\mathrm{of}\mathrm{the}\mathrm{line}\mathrm{passing}\mathrm{through}P\left(z_0\right)\mathrm{and}\mathrm{perpendicular}$$$$\mathrm{to}\mathrm{the}\mathrm{given}\mathrm{line}\mathrm{is}\mathrm{\_}\mathrm{\_}\mathrm{\_}$$

Answered by Olaf last updated on 29/Nov/20

$$\mathrm{Equation}\mathrm{of}\mathrm{the}\mathrm{straight}\mathrm{line}:$$$$\stackrel{\mathrm{\_}}{lz}+l\stackrel{\mathrm{\_}}{z}+m=0$$$$\mathrm{Equation}\mathrm{of}\mathrm{any}\mathrm{line}\mathrm{passing}\mathrm{through}$$$$\mathrm{the}\mathrm{given}\mathrm{line}:$$$$-lz+\stackrel{\mathrm{\_}}{l}\stackrel{\mathrm{\_}}{z}+q=0\left(1\right)$$$$\mathrm{For}\mathrm{a}\mathrm{perpendicular}\mathrm{line}\mathrm{passing}$$$$\mathrm{through}\mathrm{P}\left(z_0\right):$$$$-{lz}_0+\stackrel{\mathrm{\_}}{l}{\stackrel{\mathrm{\_}}{z}}_0+q=0\left(2\right)$$$$\left(2\right)-\left(1\right):l(z-z_0)-\stackrel{\mathrm{\_}}{l}(\stackrel{-}{z}-{\stackrel{\mathrm{\_}}{z}}_0)=0$$$$\Rightarrow \mathrm{Im}\left[l(z-z_0)\right]=0$$$$\Rightarrow l(z-z_0)\in \mathbb{R}$$

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