I'm faced with this problem after finding some tangent lines for the curve in the first quadrant :

I've parametrized the hyperbola by $r(t)=(t,\frac{4}{t})$, $r'(t)=(1,-\frac{-4}{t^2})$. Then by finding some tangent lines at the points $r(1)=(1,4) ,r(2)=(2,2),r(4)=(4,1)$,




After this I was told to find each triangle area that the tangent lines create when they intersect the x-axis and the y-axis.The areas that I've got are all equal to $8$. From this I'm supposed to see a hypothesis and prove it but I'm still not able to see it


The tangent in $(a,c/a)$ of $f(x)=c/x$ has the equation $$y-\frac{1}{a}=-\frac{c}{a^2}(x-a).$$ The intercepts of the tangent and the axis are $(0,2c/a)$ and $(2a,0)$.


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