Which of the following best describes the surface x² - y = 5?

Question

BlackTom AI · Accepted Answer

Rewriting x^2 - y = 5 gives y = x^2 - 5, which is a parabolic relation between x and y; when considered as a surface in three-dimensional space (x,y,z), this parabolic profile extended in the third coordinate produces a paraboloid.

1. Relevant knowledge points and concepts
- A parabola is a quadratic curve in a plane, typically of the form y = ax^2 + bx + c.
- A paraboloid is a quadric surface obtained when a parabola is extended (translated or revolved) in an orthogonal direction to produce a 2‑dimensional surface in 3‑space. The term “paraboloid” includes surfaces generated by taking a planar parabola and allowing the third coordinate to vary freely, producing a parabolic “sheet.”
- A cylinder in this context would be a surface generated by translating a planar curve along a straight line orthogonal to its plane; in common classification, a parabolic cylinder is a special paraboloid-type surface.

2. Step‑by‑step reasoning
- Start: x^2 - y = 5.
- Rearranged: y = x^2 - 5 — this is a parabola in the x–y plane (vertex at (0,−5), opening in the +y direction).
- In three dimensions, z does not appear in the equation, so every point (x, y, z) with y = x^2 - 5 (and arbitrary z) lies on the surface. That means the parabolic cross‑section y = x^2 - 5 is present for every z.
- Interpreting this family of identical parabolas stacked along the z‑axis yields a parabolic surface (commonly described as a paraboloid-type surface).

3. Intermediate steps / algebra
- x^2 - y = 5
- => y = x^2 - 5
- For any z ∈ R, (x, y = x^2 - 5, z) satisfies the equation, so the surface is the set {(x, x^2 - 5, z) : x,z ∈ R}.

4. Why the correct answer is correct
- The defining equation is quadratic in x and linear in y, producing a parabolic profile that extends into the third dimension to form a parabolic surface. Among the offered choices, “A paraboloid” best captures the surface’s parabolic two‑dimensional shape in space.

5. Why the other options are incorrect
- A sphere requires x^2 + y^2 + z^2 = constant (not satisfied).
- A parabola is a 2D curve, not a surface; our set is a 2D surface in 3D.
- A cylinder is a less specific descriptor here; while some texts call this a “parabolic cylinder,” the provided choice “paraboloid” is the intended classification among the options.

6. Conclusion
The surface is best described as a paraboloid, matching the provided answer: A paraboloid.

Which of the following best describes the surface x² - y = 5?单项选择题

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