Christian's maximum and minimum

Algebra Level 5

Let \(x\) and \(y\) be real numbers satisfying \(4x^2 + 5y^2 = 1\). Over all such pairs, let the maximum and minimum values of \(2x^2 + 3xy + 2y^2\) be \(M\) and \(N\) respectively. If \(M+N + MN = \frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?

This problem is posed by Christian L.

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