Show that \(\bar { x } = \frac { 400 - x ^ { 2 } } { 80 - 3 x }\) and find a corresponding expression for \(\bar { y }\).
The shape \(A B E F D\) is in equilibrium in a vertical plane with the edge \(D F\) resting on a smooth horizontal surface.
Find the greatest possible value of \(x\), giving your answer in the form \(\mathrm { a } + \mathrm { b } \sqrt { 2 }\), where \(a\) and \(b\) are constants to be determined.