I need to find the focii of the hyperbola that is defined by the equation $xy = 16$. I know that the focii will lie on the major axis, which is $y = x$. After doing some calculations, I found that the focii will be $16$ units away from the two vertices of the hyperbola, which are $(-4, -4)$ and $(4, 4)$.
I am stuck at finding the points that lie exactly $16$ units away from the vertices. I can partially recall a method using vectors, but I can't completely recall exactly how I should work with the vectors to get to the focii.
Any help will be greatly appreciated.
I think this is incorrect.
The intersection point of the asymptotic line $y=0$ with the tangent line at $(4,4)$ is $(8,0)$.
It follows that the distance between the center $(0,0)$ and the focus is $8$.
So, letting $(a,a)$ be the coordinates of the focus, we get $$\sqrt{(a-0)^2+(a-0)^2}=8\implies 2a^2=8^2\implies a=\pm 4\sqrt 2$$
Hence, the coordinates of the foci are $(4\sqrt 2,4\sqrt 2)$ and $(-4\sqrt 2,-4\sqrt 2)$.
Added :
Let $A(8,0),O(0,0)$ and let $F$ be the focus in the first quadrant.
Then, the above used the fact that $FO=AO.$
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