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Question Number 19592 by ajfour last updated on 13/Aug/17

Commented by Tinkutara last updated on 13/Aug/17

$$z_F=\frac{{ab}^2}{a^2+b^2}+i\frac{a^{2}b}{a^2+b^2}$$

Commented by ajfour last updated on 13/Aug/17

$$\mathrm{If}\mathrm{we}\mathrm{seek}\mathrm{to}\mathrm{find}\mathrm{equation}\mathrm{of}$$$$\mathrm{line}\mathrm{shown}\mathrm{above},\mathrm{z}={\mathrm{z}}_{\mathrm{F}}+\mathrm{i}\lambda {\mathrm{z}}_{\mathrm{F}}$$$$\mathrm{and}\overline{\mathrm{z}}={\overline{\mathrm{z}}}_{\mathrm{F}}-\mathrm{i}\lambda {\overline{\mathrm{z}}}_{\mathrm{F}}$$$$\mathrm{so}\frac{\mathrm{z}-{\mathrm{z}}_{\mathrm{F}}}{\overline{\mathrm{z}}-{\overline{\mathrm{z}}}_{\mathrm{F}}}=-\frac{{\mathrm{z}}_{\mathrm{F}}}{{\overline{\mathrm{z}}}_{\mathrm{F}}}$$$$\Rightarrow {\overline{\mathrm{z}}}_{\mathrm{F}}\mathrm{z}-\mid {\mathrm{z}}_{\mathrm{F}}{\mid }^2=-{\mathrm{z}}_{\mathrm{F}}\overline{\mathrm{z}}+\mid {\mathrm{z}}_{\mathrm{F}}{\mid }^2$$$$\Rightarrow {\overline{\mathrm{z}}}_{\mathrm{F}}\mathrm{z}+{\mathrm{z}}_{\mathrm{F}}\overline{\mathrm{z}}-2\mid {\mathrm{z}}_{\mathrm{F}}{\mid }^2=0$$$$\mathrm{if}\mathrm{equation}\mathrm{of}\mathrm{line}\mathrm{be}$$$$\overline{\alpha }\mathrm{z}+\alpha \overline{\mathrm{z}}+2\mathrm{c}=0$$$$\mathrm{then}\mathrm{this}\mathrm{means}\alpha ={\mathrm{z}}_{\mathrm{F}}\mathrm{and}\mathrm{c}=-\mid {\mathrm{z}}_{\mathrm{F}}{\mid }^2.$$$$\mathrm{complex}\mathrm{slope}=-\frac{\alpha }{\overline{\alpha }}=-\frac{{\mathrm{z}}_{\mathrm{F}}}{{\overline{\mathrm{z}}}_{\mathrm{F}}}.$$$$=-\left(\frac{\mathrm{b}+\mathrm{ia}}{\mathrm{b}-\mathrm{ia}}\right)\times \frac{\mathrm{i}}{\mathrm{i}}=\frac{\mathrm{a}-\mathrm{ib}}{\mathrm{a}+\mathrm{ib}}.$$

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