Systems of conics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:55:23Z http://mathoverflow.net/feeds/question/8202 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8202/systems-of-conics Systems of conics David Hansen 2009-12-08T16:16:34Z 2009-12-08T17:01:56Z <p>It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves intersect, they do so at a right angle. Does anyone know a good <em>elementary</em> proof of this? I.e. no complex analysis, no physics... something the ancients would've appreciated. I am looking to explain this to a sharp 13-year-old...</p> http://mathoverflow.net/questions/8202/systems-of-conics/8209#8209 Answer by Thorny for Systems of conics Thorny 2009-12-08T16:49:51Z 2009-12-08T17:01:56Z <p>As long as you can get an elementary proof of the fact that one of the families consists of ellipses with foci $A=(-c,0)$ and $B=(c,0)$ and the other consists of hyperbolas with the same foci, you can say that for any intersection point $P$ the angle between the lines $PA$ and $PB$ is dissected by the tangent to either curve - otherwise moving on the tangent would cause a first-order error in the sum $|PA|+|PB|$ or the difference $|PA|-|PB|$, respectively. Hence the two tangents are just the two angle bisectors of a pair of lines, and are thus orthogonal.</p>