2022 AMC 12A Problem 22

Let c be a real number, and let z_1 and z_2 be the two complex numbers satisfying the equation z^2 - cz + 10 = 0. Points z_1, z_2, \frac{1}{z_1}, and \frac{1}{z_2} are the vertices of (convex) quadrilateral \mathcal{Q} in the complex plane. When the area of \mathcal{Q} obtains its maximum possible value, c is closest to which of the following?

\textbf{(A) }4.5 \qquad\textbf{(B) }5 \qquad\textbf{(C) }5.5 \qquad\textbf{(D) }6\qquad\textbf{(E) }6.5

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