You know that there's two equations to be solved simultaneously

.... (1)

.... (2)

and three unknowns x, y and z. So clearly there's gonna be a parameter. Choose the parameter to be real convenient. is an obvious choice (see below *) because when you substitute it into (2) you get:

.... (3),

a nice easy-on-the-eye equation.

Solve (1) and (3) simultaneously for x and y in terms of t:

Substitute (1) into (3):

.

Expand, simplify and solve for x. . Now solve for y.

The curve is the intersection of the infinite cylinder and the sphere . I leave the visualisation of this to you.

* You should see what solution you get if you choose ......