I am having problems solving this.
Find a vector tangent to the curve of intersection of $z = 4x^2 + y^2$ and $z=(27-x^2-y^2)^{1/2}$ at the point $(1,1,5)$.
I'm able to do this kind of thing using parametrization and vector functions when the equation is much simpler, but this seems too complex. I used wolfram alpha to determine the projection onto $x-y$ plane is an ellipse... that's about it.