AHSME 1960 Problem 22

The equality $(x+m)^2-(x+n)^2=(m-n)^2$, where $m$ and $n$ are unequal non-zero constants, is satisfied by $x=am+bn$, where: $\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value} \\ \textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values} \\ \textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value} \\ \textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values} \\ \textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value}$