Conic Complex

Geometry Level 4

f(x,y)=nx2+(nm2)xy+my2+4x+2y+1=0g(x,y)=nx2+(n+m)xy+my2+4x+2y+1=0h(x,y)=nx2+(nm)xy+my2+4x+2y+1=0\begin{aligned} 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{aligned}

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

If nn and mm are positive integers satisfying the constraints above, compute 2n+m2n + m.

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