AHSME 1967 Problem 13

A triangle $ABC$ is to be constructed given a side $a$ (opposite angle $A$). angle $B$, and $h_c$, the altitude from $C$. If $N$ is the number of non-congruent solutions, then $N$ $\textbf{(A)}\ \text{is} 1 \textbf{(B)}\ \text{is} 2 \textbf{(C)}\ \text{must be zero} \textbf{(D)}\ \text{must be infinite} \textbf{(E)}\ \text{must be zero or infinite}$