Think Geometry, Not Algebra

Geometry Level 5

\[\begin{eqnarray} &&2\sqrt{(x^{2}-8x+y^{2}+16)(x^{2}-6x+y^{2}-14y+58)} \\ && +2x^{2}-14x+2y^{2}-14y+74 \end{eqnarray} \]

For real numbers \(x,y\), what is the minimum value of the above expression, subject to the constraint \(3x+y-6 = 0\)?


Inspired by my slack conversation with Manisha Garg.
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