Let there be two 2nd degree curves: $$f(x,y)=ax^2+by^2+cx+dy+e=0$$ and $$g(x,y)=fx^2+gy^2+hx+iy+j=0,$$ how is it possible to determine if these two curves intersect in some region, say $x \le 1 , y \ge 1$, without actually calculating the roots of these two curves.
Alternatively what are the conditions on the coefficients for these two curves to intersect in the region $x \le 1 , y \ge 1$.
Any help would be appreciable
This is not what you want:
It is hard to say what concrete conditions on the coefficients such that $F,G$ intersect in $\mathbb{C}^2$.
However, by Chevalley's theorem, we know some informations of the set of coefficients such that $F,G$ intersect in $\mathbb{C}^2$.
Consider the map $\mathbb{C}[a,b,c,d,e,f,g,h,i,j]\to \mathbb{C}[a,b,c,d,e,f,g,h,i,j][x,y]/(F,G)$. Let $X,Y$ denote the affine schemes respectively corresponding to the rings. So we have a map from $Y\to X$, by Chevalley's theorem, we know that the image of $Y$ is a constructible set of $X$.
If we only consider the closed points $X_0$ of $X$ and $Y_0$ of $Y$. Then $F,G$ intersect if and only if $(a,b,c,d,e,f,g,h,i,j)$ in the image of $Y_0$.