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.