calculus 2 - surfaces and Implicit function question (homework)

Question

I am stuck hard on this one: find all constants C so that the surfaces: $$X^2+Y^2+Z^2=1$$ $$Z=X^2+Y^2+C $$ 1)Tangent at the common points

2)perpendicular at the common points.

Any help would be really appreciated.

Answer 1

Define $f(x,y,z) = x^2 +y^2 +z^2 -1$ and $g(x,y,z) = z -x^2 -y^2 -c$. Then you have your pair of surfaces for $f(x,y,z) = 0$ and $g(x,y,z) =0$.

Steps for part 1:

1. If they are tangent at common points then their normal vectors are collinear. Using gradients you have $\nabla f = \lambda \nabla g$. Use this to find a condition for $\lambda$.
2. This will give you $z$. Solve the system $$\begin{cases} x^2+y^2+z^2 & = 1, \\ z -x^2-y^2 & = c, \end{cases}$$ to find the value of $c$.

Steps for part 2:

1. If they are orthogonal at common points then their normal vectors are orthogonal. Using gradients you have $\nabla f \cdot \nabla g = 0$. Use this to find an equation relating $z, x$ and $y$.
2. Use the same system to find the value of $c$.