The equations of two lines making an angle 45° with a given line $\mathrm{a}\stackrel{\mathrm{\_}}{\mathrm{z}}+\mathrm{a}\stackrel{\mathrm{\_}}{\mathrm{z}}+\mathrm{b}=0$ (where ‘a’ is a complex number and b is real) and passing through a given point C (c) (c is a complex number), is/are

$\mathrm{Let} \mathrm{z} \mathrm{be} \mathrm{the} \mathrm{locus} \mathrm{point} \mathrm{on} \mathrm{the} \mathrm{required} \mathrm{line}$

${\mathrm{z}}_1,{\mathrm{z}}_2 \mathrm{be} \mathrm{two} \mathrm{points} \mathrm{on} \mathrm{the} \mathrm{given} \mathrm{line}$$\mathrm{a}\stackrel{\mathrm{\_}}{\mathrm{z}}+\mathrm{a}\stackrel{\mathrm{\_}}{\mathrm{z}}+\mathrm{b}=0$

using coni method

$\frac{\mathrm{z}-\mathrm{c}}{|\mathrm{z}-\mathrm{c}|}=\frac{{\mathrm{z}}_1-{\mathrm{z}}_2}{\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}{\mathrm{e}}^{\pm \mathrm{i}\frac{\mathrm{\pi }}{4}}$

squaring on both sides

$\frac{{\left(z-c\right)}^2}{\left(z-c\right)\left(\overline{z}-\overline{c}\right)}=\frac{{\left({\mathrm{z}}_1-{\mathrm{z}}_2\right)}^2}{\left({\mathrm{z}}_1-{\mathrm{z}}_2\right)\left(\overline{{\mathrm{z}}_1}-\overline{{\mathrm{z}}_2}\right)}{\mathrm{e}}^{\pm \mathrm{i}\frac{\mathrm{\pi }}{2}}$

$\frac{\mathrm{z}-\mathrm{c}}{\overline{z}-\overline{\mathrm{c}}}=\frac{{\mathrm{z}}_1-{\mathrm{z}}_2}{{\overline{\mathrm{z}}}_1-{\overline{\mathrm{z}}}_2}{\mathrm{e}}^{\pm \mathrm{i}\frac{\mathrm{\pi }}{2}}$

$\frac{\mathrm{z}-\mathrm{c}}{\overline{\mathrm{z}}-\overline{\mathrm{c}}}=\frac{-\mathrm{a}}{\overline{\mathrm{a}}}(\pm \mathrm{i})$

$\frac{\mathrm{z}-\mathrm{c}}{\mathrm{a}}\pm \mathrm{i}\frac{\overline{\mathrm{z}}-\overline{\mathrm{c}}}{\overline{\mathrm{a}}}=0$