2007 AMC 10B Problem 10

Two points B and C are in a plane. Let S be the set of all points A in the plane for which \triangle ABC has area 1. Which of the following describes S?

\textbf{(A) } \text{two parallel lines} \qquad\textbf{(B) } \text{a parabola} \qquad\textbf{(C) } \text{a circle} \qquad\textbf{(D) } \text{a line segment} \qquad\textbf{(E) } \text{two points}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.