Common chord between 2 different conics

Question

Suppose I have 2 different conics - for example, a circle and a parabola. How do I find the common chord between them? I tried implementing the $S_1-S_2=0$ approach, but it is not giving me any proper answers, as it results in equations like $x^2+6x-4y=0$, which I have no idea how to obtain the common chord from.

Answer 1

Hint...try $$S_1-\lambda S_2=0$$ and choose $\lambda$ so the non-linear terms vanish