Hello, catcat103!
1. Find the points of contact between surfaces: .
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
. . and a hyperbola through the point
They intersect at
The other branch of the hyperbola intersects the circle at
The sphere and hyperbolic cylinder intersect at: