$12 \cdot 29$ If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a
(A) right triangle.
(B) circle.
(C) hyperbola.
(D) line.
(E) parabola.
(42nd American High School Mathematics Examination, 1991)
[Solution] Let $z=x+y i$, then we have
$$
(3+4 i)(x+y i)=(3 x-4 y)+(4 x+3 y) i
$$

to be a real number, then $4 x+3 y=0$, which is a straight line. Therefore, the answer is $(D)$.