Hyperbola + Locus = Deadly Combination!

Geometry Level pending

A variable straight line of slope $$4$$ intersects the hyperbola $$xy=1$$ at two points. The locus of the point which divides the segment between these points in ratio $$1:2$$ can be written as $$ax^2+2hxy+by^2+2gx+2fy=c$$ where $$a,h,b,g,f,c\geq 0$$ and $$\gcd(a,b,c,f,g,h)=1$$.

Find $$a+b+c+f+g+h$$.

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