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.