Find the locus of $z$ satisfying $|z - ia| = \text{Im}(z) + 1$, where $a$ is a real constant.
Let $z = h + ik$. Then $\sqrt{h^2 + (k-a)^2} = k+1$, giving $h^2 - 2k(a+1) + (a^2-1) = 0$. The locus is $x^2 - 2y(a+1) + a^2 - 1 = 0$, which is a parabola. (A)
Find the locus of $z$ satisfying $|z - ia| = \text{Im}(z) + 1$, where $a$ is a real constant.