Straight Lines in Complex Plane/ Inversion Function

Solution 1:

Suppose that $w$ is a fixed complex number. $\mathbb{C}$ has an analogue of a dot product, thinking of it as a vector space over itself. This Hermitian inner product is written as $\langle z, w \rangle = \bar{z}w $. This is clearly $0$ if and only if one of the arguments is $0$. However, writing $z = a + bi, w = c + di$, we see $\Re(\bar{z}w) = ac + bd = 0$ when $ac = -bd$, which is the same as $a/b = -d/c$, i.e. these complex numbers are perpendicular as vectors in the real plane. In fact, you can check that $\Re{(\bar{z}}w)$ is nothing more than the familiar inner product on the plane. So for a fixed $z$ or $w$, this is saying to take all vectors that have a given inner product against that fixed vector. The locus of such vectors is a line, being defined by linear equations.