Question

On which of the following types of quadric surface does the following parametrized curve

r(t)=⟨tsin(t),3t²,-tcos(t))
lie?
A. cone
B. sphere
C. ellipsoid, but not a sphere
D. paraboloid
E. None of the above.

Answer 1

The parametrized curve r(t) lies on a paraboloid.

Option D is answer.

Step-by-step explanation:

To identify the type of quadric surface, we need to examine the equation for the curve. Here, we see that the x-component (tsin(t)) is squared, while the y-component (3t^2) is not. This indicates that the surface is not symmetric along the x-axis, eliminating options A (cone) and C (ellipsoid).

Furthermore, the z-component (-tcos(t)) is also squared, leading to a hyperbolic shape. This eliminates option D (paraboloid) as paraboloids have parabolic sections along one axis and circular sections along the other.

Therefore, the only option remaining is E. None of the above. The curve r(t) represents a hyperbolic paraboloid, which is not a standard quadric surface like sphere, ellipsoid, etc.

Option D is answer.