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