Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 1 / 8A / ii / Solution

The quotient is defined only when $z+\overline{u}\ne 0$. Its unit-modulus condition then gives $|z+u{|}^2=|z+\overline{u}{|}^2$. On expanding with complex conjugates,Hence at least one imaginary part is zero, and either $z$ or $u$ is real. Conversely, either reality condition makes the two squared moduli equal; provided the denominator is nonzero, the quotient has modulus one. No conclusion is drawn from an undefined quotient.