Compute the points of the intersection between the circle x^2+y^2=1 and hyperbola xy=1?
I tried to put 1/y in place of x but I couldn't solve.
You have $\displaystyle x^2+y^2=1$ and $\displaystyle xy=1$.
We may assume $\displaystyle x,y\not=0$ since they don't lie on the hyperbola anyways.
Taking $\displaystyle x=1/y$ as you have tried,
$\displaystyle \frac{1}{y^2}+y^2=1$
Multiplying by $\displaystyle y^2$ and rearranging gives
$\displaystyle y^4-y^2+1=0$
which has no real solutions. So there is no intersection. In fact, this is obvious if you graphed the two functions.
Yes, but the discriminant is negative, so all the roots are non-real numbers in the complex plane. That is to say there is no real solution. As I have suggested earlier, just graphing these two functions makes it clear that they do not intersect on the xy plane.