In this method, the equations of the ellipse and the circle are analyzed together to determine their intersection points analytically. The circle equation is first transformed into its parametric form and then substituted into the ellipse equation. This results in a fourth-degree polynomial. The real roots of the polynomial provide the parameter values corresponding to the intersection points. These parameter values are then used to calculate the coordinates of the points where the circle and the ellipse intersect.
General equation of the ellipse:
\begin{equation}
\label{elips}
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
\end{equation}
General equation of the circle:
\begin{equation}
\label{cember}
(x-a)^2 + (y-b)^2 = r^2
\end{equation}
Parametric form of the circle:
\begin{equation}
\label{x}
x = a + r\frac{1-t^2}{1+t^2}
\end{equation}
\begin{equation}
\label{y}
y = b + r\frac{2t}{1+t^2}
\end{equation}
Equations \ref{x} and \ref{y} are substituted into the ellipse equation \ref{elips}, resulting in a fourth-degree equation. The real roots of this
quartic equation give the corresponding values of the parameter \(t\).
Using the values of \(t\) obtained from the quartic equation, the corresponding coordinates \(x\) and \(y\) are calculated using the formulas above.
The resulting coordinates are the intersection points of the circle and the ellipse.