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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 $r(1)=(1,4) ,r(2)=(2,2),r(4)=(4,1)$,

$l_1:r_1(t)=(1,4)+t(1,-4)$

$l_2:r_2(t)=(2,2)+t(1,-1)$

$l_3:r_3(t)=(4,1)+t(1,-\frac{1}{4})$

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

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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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