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
$\fbox{D}$
Solution 1:
We want $(3+4i)z$ a real number, so we want the $4i$ term to be canceled out. Then, we can make $z$ be in the form $(n-\frac{4}{3}ni)$ to make sure the imaginary terms cancel out when it's multiplied together. $(n-\frac{4}{3}ni)$ is a line, so the answer is $\textbf{(D) } line$