It seems wellknown that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2c^2}=1$ 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 elementary 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 13yearold...
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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 firstorder error in the sum $PA+PB$ or the difference $PAPB$, respectively. Hence the two tangents are just the two angle bisectors of a pair of lines, and are thus orthogonal. 

