This is from the book An introduction to differential geometry by Willmore.
What function $f(u)$ would make the curve $\vec r=(a\cos u,a \sin u, f(u))$ a plane?
My solution: If for example, I take $z=-a\cos u-a\sin u+c$ (where $c$ is a constant) then $x+y+z=c$ which is an equation of a plane.
Is this correct and if so are there any other solutions?
Any $a x+b y+c z= d$ is a plane, and so $$f(u)= d+ \lambda a \cos u + \mu a \sin u$$ is the function you are looking for.