A 'nice' form of writing a line through $2$ points $z_1,z_2\in\mathbb{C}$ is $$z\overline{z_1-z_2}+z_1\overline{z_2-z}+z_2\overline{z-z_1}=0$$ Now I'm asked to give a similar expression for a general conic. Now my idea was to just 'translate' the definition of a conic to complex numbers as follows:
$$(S\cdot z+T-1)\overline{S\cdot z+T-1}-e^2(S\cdot z+T+\overline{S\cdot z+T})^2=0$$ Which is based on using the point $(0,1)$ as the focus point, and the imaginary axis as the directrix, and then just translating and scaling/rotating using $S$ and $T$.
However, this form does not look as 'elegant' as the expression for the line. So does anyone know of a nicer way of writing the form of a general conic?
Thanks.
Edit: I think this form is also correct, and it's already a bit more elegant
$$S(z-F)\overline{S(z-F)}-e^2(Sz+\overline{Sz})^2=|S(z-F)|-e^2(Sz+\overline{Sz})^2=c$$