Let the function g be defined on the complex numbers as g(z) = (a + 2 i)z for some real value of a.
Then if g(z) is equidistant from 0 and z for all z, what is a equal to?
Since g(z) is equidistance from 0 and z, |g(z)| = |g(z) + z|. This leads to a = -4.
