The equation of the line joining the complex numbers $-5 + 4i$ and $7 + 2i$ can be expressed in the form $az + b \overline{z} = 38$ for some complex numbers $a$ and $b$. Find the product $ab$.
I don't understand how I can express this in the form $az + b \overline{z} = 38$. I know that the equation of a line in the complex plane is $z = u + t(v-u)$ where t is any real number.
Best Answer
Forget complex numbers for a moment, and write down an equation in terms of $x,y$: the usual thing with point and slope. Then plug $$x=\frac12 (z+\bar z),\quad y=\frac{1}{2i}(z-\bar z)$$ to convert the equation into $z$ and $\bar z$.
Finally, multiply the equation by a constant to make the right hand side $38$.