Suppose that we have algebraic equations for a circle and a hyperbola given by $$x^2 + y^2 = 1$$ $$Ax^2 + 2Bxy - Ay^2 + Cx + Dy = 0$$ respectively. The real coefficients $A,B,C,D$ have a negative discriminant $\Delta = -(A^2 + B^2)$. These two curves intersect due to the origin being a root to the hyperbola equation, which can be viewed as an unbounded function in the $xy$-plane.
Does there exist algebraic geometric methods for finding the intersection points? I have little knowledge of the subject but if there are methods for specifying the intersection points of these simple curves I would like to know. I know that substitution leads to a fourth order polynomial but solution by radicals leads to complicated formula which seem intractable. I am interested in finding these intersection points as they correspond to equilibrium points of a system of differential equations. I am curious if anyone knows a change of coordinates that make finding these intersections a little more tractable.