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Activity 10.1.4.
For this activity we will consider the paraboloid with equation
\(z = x^2+y^2\text{.}\)
(a)
Parameterize the intersection of the paraboloid with the fundamental plane
\(x=2\text{.}\) What type of curve is the intersection?
(b)
Parameterize the intersection of the paraboloid with the fundamental plane
\(y = -1\text{.}\) What type of curve is the intersection?
(c)
Parameterize the intersection of the paraboloid with the fundamental plane
\(z = 25\text{.}\) What type of curve is the intersection?
(d)
How do your responses change to all three of the preceding questions if you instead consider the surface defined by
\(z = x^2 - y^2\text{?}\)
Hint.
For generating one of the parameterizations you may want to consider
\(\sec^2(t)-\tan^2(t) = 1\text{.}\)
