Equation in Progressions

Algebra Level 2

The harmonic mean of two numbers \(a\) and \(b\) is 4. It is also given that \(2A + G^2 = 27\), where \(A\) and \(G\) are the arithmetic mean and geometric mean of \(a\) and \(b\) respectively.

Find the numbers \(a\) and \(b\).