Optimisation Problem

Hey, I really liked your response to the last question I asked.

Could you please explain what the person means by "vanishing determinant condition," and how he derives that $h=2\sqrt{(2r+R)(r+2R)} $ 

and also please explain the second case because I don't understand why he says "return to the cartesian system" and how he gets those equations. 

Here's the website again: https://math.stackexchange.com/questions/4024363/optimizing-a-conical-frustum-using-partial-differentiation/4024472#4024472
Thanks a lot!


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