1. The problem statement, all variables and given/known data Find all points for which the curves [itex]x^2+y^2+z^2=3[/itex] and [itex]x^3+y^3+z^3=3[/itex] share the same tangent line. 2. Relevant equations Sharing the same tangent line amounts to having the same derivative. The constraint then is that [itex]3x^2+3y^2+3z^2=2x+2y+2z[/itex]. The points must obviously also lie on the original curves. 3. The attempt at a solution Combining the constraint on the derivatives ([itex]3(x^2+y^2+z^2)-2(x+y+z)=0[/itex]) with the constraint that [itex]x^2+y^2+z^2=x^3+y^3+z^3=3[/itex] we see that the constraint on the derivatives becomes [itex]3(3)-2(x+y+z)=0[/itex] which is just the planar equation [itex]2(x+y+z)=9[/itex]. This feels wrong to me; these curves should not intersect at a plane. Am I right?