Prove that the center of the ellipse $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$, where $A, B, C, D, E>0$, is not in the first quadrant of the $xy$-plane?

Question

Suppose that for $A, B, C, D, E>0$, the quadratic curve \begin{equation} Ax^2+Bxy+Cy^2+Dx+Ey+F=0 \end{equation} is an ellipse. Could we conclude that the center of the ellipse is NOT in the first quadrant of the $xy$-plane? I have plotted several examples, which suggest that the answer is affirmative. Any reference, suggestion, idea, or comment is welcome.

Answer 1

When $4AC - B^2 > 0,$ it is an ellipse or single point or empty. The center is the point where $$ 2Ax + By + D = 0,$$ $$ B x + 2 C y + E = 0 $$ and cannot be in the first quadrant