a) The equation of the line through $2 + 3i$ and $0$ can be written as $$az + b \overline{z} = 0$$ for some complex numbers $a$ and $b$. What is the quotient $b/a$ in rectangular form. Should I graph it out?
b) The equation of the line through $2+3i$ which is perpendicular to the line through $0$ and $2+3i$ has equation $$az + b\overline{z} = 26,$$ where $a$ and $b$ are constant complex numbers. Find the product $ab$ in rectangular form. Same concept... I am confused on how to approach.
(a) since line pass through affix 0 and affix $z_{1}=2+3i$ so, it's equation will be
$z\ \overline {z_{1}}-\overline {z_{}} \ z_{1}=0\implies a=\overline {z_{1}}=2-3i=\sqrt{13}\ e^{-i{{\theta_{1}}}}\ \ $ and $\\b=z_{1}=2+3i=\sqrt{13}\ e^{i\theta_{1}}\implies\dfrac{b}{a}=e^{2i\theta_{1}}$ where $\theta _{1}=tan ^{-1}\frac{3}{2}$
(b) equation of line $\perp$ to line (a) and which also passes through $2+3i$ (and of course through any point $z$ ) will be
$[0-(2+3i)](2-3i-\overline z)+ [(0-(2-3i))](2+3i-z)=0\implies(2+3i)\overline z +(2-3i)z=26$
hence ,
$ a= (2-3i)\ \ $ and $\ b=2+3 i \implies ab=13$