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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