Hello, yzc717;230977!

I would use Lagrange Multipliers on #4 . . .

4. Find the points on the curve of intersection of the surfaces: .

in that are closest to the origin.

We want to minimize: .

. with the constraints: .

We have: .

Take partial derivatives and equate to zero . . .

. .

. .

. .

. .

. .

Then solve the system for: .