Bizarre Hyperbola

Calculus Level 4

The hyperbola above has equation 2x2+10xy+8y2+4x+2y+1=02x^2 + 10xy + 8y^2 + 4x + 2y + 1 = 0.

Let m1m_{1} and m2m_{2} be the slopes of the asymptotes (shown in green) such that m1>m2|m_{1}| > |m_{2}| , and let m3m_{3} be the slope of the line that passes through both points, where the tangents have infinite slopes.

Find the value of 1m1m2m3\dfrac{1}{|m_{1}| - |m_{2}| - |m_{3}|}.

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