I am looking for an algorithm for finding the intersection of a line and an ellipse.
I have the line in the form
$ax+by+c=0 \qquad(1)$
and the ellipse in the form
$Ax^2+Bxy+Cy^2+Dx+Ey+F=0 \qquad(2)$
My idea is to solve in a closed form the system $(1),(2)$. Is it too naive? What are the pitfalls of such approach?