Which surface is created by the intersection of the cylinder and the plane
That is, the cylinder cuts out a surface from the plane.
Drawing a sketch is very helpful (in this case, it is not very difficult to do).
First we choose a coordinate system in the plane , i.e. an origin and an orthonormal basis , so we can write the equation of the intersection using it. For instance (as shows the sketch, it is not only a simple but also an appropriate choice), , and .
Then, for any in the intersection, , hence , so the coordinates of in the new system are and . The equation becomes . In other words, the intersection is an ellipse with minor and major axes respectively equal to and , centered at . The major axis is in direction and the minor one in direction .