The cylinder is a clue to use cylindrical coordinates. Taking 
and 
, we then are forced to use 
. So the parameterization of the intersection of the plane and cylinder is
To get the surface, we can introduce a second parameter
that "contracts" the elliptical intersection to a point. The simplest way to do this is to use
with
and 
.
To get the surface, we can introduce a second parameter
with