Minimum value too dangerous

Algebra Level 5

\[433x^{2}+y^{2}+\sqrt{1729}xy+\sqrt{3}x-\sqrt{3}y+1729!\] For real values of \(x,y\) the above expression attains minimum value at \(x=M,y=N\) . And \(M+N=a\sqrt{b}\) where \(a\) is positive integer and \(b\) is square root free. Find \(a-b\).

Problem is original.
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