Activity 10.2.4

For this activity we will consider the paraboloid with equation $z = x^2+y^2$. You do not need to know what the graph of $z=x^2+y^2$ looks like at this point.

1. Parameterize the intersection of the paraboloid with the fundamental plane $x=2$. What type of curve is the intersection?

2. Parameterize the intersection of the paraboloid with the fundamental plane $y = -1$. What type of curve is the intersection?

3. Parameterize the intersection of the paraboloid with the fundamental plane $z = 25$. What type of curve is the intersection?

4. How do your responses change to all three of the preceding questions if you instead consider the surface defined by $z = x^2 - y^2$?