Calculating an axis aligned bounding box for a transformed sphere

Question

I'm trying to understand the math behind this answer :

https://stackoverflow.com/questions/4368961/calculating-an-aabb-for-a-transformed-sphere/4369956#4369956

I've implemented it and it seems to work, but I don't know why ..

Can somebody point me to a definition of the dual of a conic and why it allows us to find the planes tangent to the tranformation ?

Thanks.