# Conic Complex

Geometry Level 4

\begin{align} 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 \end{align}

Let the graphs $$f, g, h$$ be a parabola, a hyperbola, and an ellipse, respectively.

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

×