How to tell if a second-order curve goes below the $x$ axis? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T23:40:32Z http://mathoverflow.net/feeds/question/121485 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121485/how-to-tell-if-a-second-order-curve-goes-below-the-x-axis How to tell if a second-order curve goes below the $x$ axis? Felix Goldberg 2013-02-11T15:06:42Z 2013-02-11T18:15:33Z <p>Suppose we have a second-order curve in general form:</p> <p>(1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$.</p> <p>I'd like to know if there is a simple condition that ensures that the curve has at least one point on on or below the $x$ axis, <em>i.e.</em> that the left-hand side of (1) is nonpositive.</p> <p>In the trivial case that the curve is a parabola, the discriminant being nonnegative is just such a condition. But what happens in the general case?</p> http://mathoverflow.net/questions/121485/how-to-tell-if-a-second-order-curve-goes-below-the-x-axis/121492#121492 Answer by Peter Michor for How to tell if a second-order curve goes below the $x$ axis? Peter Michor 2013-02-11T15:54:11Z 2013-02-11T15:54:11Z <p>Solve for $y$ in the form $y= A(x) \pm \sqrt{B(x)}$ and estimate. More abstract versions are just variant of this. </p> http://mathoverflow.net/questions/121485/how-to-tell-if-a-second-order-curve-goes-below-the-x-axis/121511#121511 Answer by Serge Lvovski for How to tell if a second-order curve goes below the $x$ axis? Serge Lvovski 2013-02-11T18:15:33Z 2013-02-11T18:15:33Z <p>We may regard the left-hand side of the equation of thecurve as a quadratic polynomial in $x$. If $D(y)$ is its discriminant (with respect to $x$), then $D(y)\ge 0$ iff there exists a point with the second coordinate $y$ on the curve. Solve this inequality for $y$ and check whether its minimal solution is negative:)))</p>