If by eliminating $x$ between the equations $x^2+ax+b=0$ and $xy+l(x+y)+m=0$ then a quadratic in $y$ is formed whose roots are the same as the original quadratic in $x$. Then prove that either $a=2l$ and $b=m$ or $b+m=al$.
It seems a very lengthy process to eliminate $x$. Does anyone know of any shorter method?