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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

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    Translate your curves so that the region boundaries align with coord axes; comparing coords to $0$ is easier. Use Sylvester's Eliminant (or resultants) to eliminate $y$ from your system and get a quartic $p(x)$ whose roots are the $x$-coords of the intersections; likewise, get $q(y)$. Then, the Descartes Rule of Signs can give insights into how many roots of $p(x)$ and $q(y)$ lie on either side of $0$. Of course, Sylvester is cumbersome, and Descartes often inconclusive --and not knowing how the $p$-roots pair-up with $q$-roots is problematic-- but this approach *could* help in certain cases.2012-05-01

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