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 \(a+b+c+f+g+h\).


Source: JEE advanced 1997.
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