Hello, catcat103!
1. Find the points of contact between surfaces: .$\displaystyle \begin{array}{c}x^2+y^2+z^2\:=\:8 \\ xy\:=\:4 \end{array}$
where they are tangential to each other.
I made a sketch and "eyeballed" the answer . . . Code:

 o

 o
* * * o
*  * (2,2)
*  ◊
*  * o
 o
*  * o
  *     *     *          
*  *

*  *
*  *
*  *
* * *

Looking at the xyplane we have a circle with radius $\displaystyle 2\sqrt{2}$
. . and a hyperbola through the point $\displaystyle (2,2).$
They intersect at $\displaystyle (2,2).$
The other branch of the hyperbola intersects the circle at $\displaystyle (2,2).$
The sphere and hyperbolic cylinder intersect at: $\displaystyle (2,2,0)\text{ and }(2,2,0)$