 Parity and upper bound for transverse ellipse intersections

Parametrize one ellipse as $\gamma (t)=c+A(\cos t,\sin t)$ and substitute it into the quadratic equation $Q=0$ of the other. At a transverse intersection, $g(t)=Q(\gamma (t))$ has a simple zero, so its sign changes. Periodicity makes the number of zeros even. The function is a trigonometric polynomial of degree at most two, so multiplication by $z^2$, with $z=e^{it}$, gives a nonzero polynomial of degree at most four. Hence there are at most four distinct intersection points. Coincident ellipses are excluded by transversality.