Finding where two curves share tangent lines

[slamminsammya]

Homework Statement

Find all points for which the curves $x^2+y^2+z^2=3$ and $x^3+y^3+z^3=3$ share the same tangent line.

Homework Equations

Sharing the same tangent line amounts to having the same derivative. The constraint then is that $3x^2+3y^2+3z^2=2x+2y+2z$. The points must obviously also lie on the original curves.

The Attempt at a Solution

Combining the constraint on the derivatives ($3(x^2+y^2+z^2)-2(x+y+z)=0$) with the constraint that $x^2+y^2+z^2=x^3+y^3+z^3=3$ we see that the constraint on the derivatives becomes $3(3)-2(x+y+z)=0$ which is just the planar equation $2(x+y+z)=9$. This feels wrong to me; these curves should not intersect at a plane. Am I right?

 

[LCKurtz]

Those aren't curves, they are surfaces. Tangent lines aren't usually what you talk about with surfaces although I guess there's no law against it.

If it's any help to you, here's a picture of the two surfaces:

 

[dynamicsolo]

I believe this is not quite finished, though. You have found that your points must lie in the plane x + y + z = 9/2 , but you must still introduce a constraint, since plainly this equation alone permits "solutions" which are far from the sphere. You could, say, write z in terms of x and y using the equation for the sphere.