Conic Complex

Geometry Level 4

\[f(x , y) = nx^2 + \left(\dfrac{nm}{2}\right)xy + my^2 + 4x + 2y + 1 = 0\]

\[g(x , y) = nx^2 + (n + m)xy + my^2 + 4x + 2y + 1 = 0\]

\[h(x , y) = nx^2 + (n - m)xy + my^2 + 4x + 2y + 1 = 0\]

Let the graphs \(f, g, h\) be a parabola, hyperbola, and ellipse respectively.

If \(n\) and \(m\) are positive integers satisfying the constraints above, compute \(2n + m\).

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