I couldn't find a clever way so this is just straightforward.
We have a system of two conic equations. Find the solutions and pick the one that minimizes xy.
Solve both equations for y and substitute to get 1/(12x)=3/81/x which simplifies to the quadratic 0=x^212x+32=(x4)(x8)
The solutions to the system (x,y) and their product:
(4,1/8) and xy=1/2
(8,1/4) and xy=2
So the smallest xy is 1/2.

Posted by Jer
on 20190129 08:53:08 