Christian's maximum and minimum

Algebra Level 5

Let xx and yy be real numbers satisfying 4x2+5y2=14x^2 + 5y^2 = 1. Over all such pairs, let the maximum and minimum values of 2x2+3xy+2y22x^2 + 3xy + 2y^2 be MM and NN respectively. If M+N+MN=abM+N + MN = \frac{a}{b}, where aa and bb are coprime positive integers, what is the value of a+ba+b?

This problem is posed by Christian L.

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