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Expressing the equation of the given ellipse in standard form.
\begin{equation}
\label{ellipse}
Ax^2+Bxy+Cy^2+Dx+Ey+F=0
\end{equation}
Expressing the equation of the given circle in standard form.
\begin{equation}
\label{cember}
\left ( x-a \right )^2+\left ( y-b \right )^2=r^2
\end{equation}
Parametric transformation 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{ellipse}, obtaining a fourth degree equation. The real number roots of this fourth degree equation give the value of the intersection \(t\).
From the \(t\) values found in the fourth degree equation, \(x\) and \(y\) are found with the above formula.
The result gives us the intersection point of the ellipse and the circle.
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