AMC 12 2019b Problem 6

In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?

$\textbf{(A) }0 \textbf{(B) }2 \textbf{(C) }4 \textbf{(D) }8 \textbf{(E) }\text{infinitely many}$