The surface with equation $z = x^2+ y^2$ is a parabolic bowl sitting on the origin. If I cut it with a plane perpendicular to the z-axis, I get a circle. If I cut it with a plane parallel to the z-axis, I get a parabola. Find the plane which is tangent to the surface at the point (1, 2, 5).
Now this looks like a fine problem in a multivariable calculus course, and indeed it is. But here I’m looking for an algebraic solution. There are actually lots of ways to do it, but here’s an idea: the tangent plane will have only one intersection with the bowl––at the point (1, 2, 5).
My professor always includes these questions at the end of his excercises, ones that are meant to challenge us, but they're always so intimidating. Can someone help me through the thought process to this question?