# Finding the Focii of a Rotated Hyperbola

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 found that the focii will be $$16$$ units away from the two vertices of the hyperbola

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 $$(-4\sqrt 2,-4\sqrt 2)$$.

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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