$\because$ , if the set $S$ consists of all points $z$ such that $(3+4 \mathrm{i}) z$ is a real number, then the set $S$ is a ( )
(A) Parabola
(B) Hyperbola
(C) Line
(D) Circle
5. (C).

The problem states that $z=a+b \mathrm{i}$, thus $(3+4 \mathrm{i})(a+b \mathrm{i})=A$, where $A$ is a real number, then $3 a-4 b=A, 4 a+3 b=0$, so the set $S$ is a straight line.