Find all tangent lines to the graph of $f(x,y) = \frac{1}{\sqrt {x^2 + y^2}}$ that are parallel to the line $x = y = z$
Edit: Some additional information:
That is everything I was given in the file with the additional problems for my Calculus 2 course.
When I asked the lecturer about the problem, he said that I should consider substituting x and y with something of t, but that advice was given a week ago and I've already forgotten all the details.
Before I stuck with the problem, I figured out the direction of the partial derivatives which is $u = (\frac{\sqrt 2}{2}, \frac{\sqrt 2}{2})$, the gradient (and partial derivatives) $\nabla f(x,y) = (\frac{-x}{(x^2 + y^2)^{\frac{3}{2}}},\frac{-y}{(x^2 + y^2)^{\frac{3}{2}}})$, that all partial derivatives in the direction of $u$ should have the value of the slope of the line $x = y = z$ which, I believe, is $\frac{\sqrt 3}{3}$ and that I need to find all points on the graph of $f(x,y)$ that give such slope and express them with a set builder and with know how to find any of those points I can make a set builder for all the lines that are tangent to the graph and are parallel to the line.