Quadric surfaces in standard form
Problem 9.903 · easy
Write \( \displaystyle 3600 x^{2} + 2025 y^{2} - 1296 z^{2} = 32400 \) in standard form and identify the surface.
- Divide through by 32400.Reviewed
- \[ \frac{3600 x^{2} + 2025 y^{2} - 1296 z^{2}}{32400} = \frac{x^{2}}{9} + \frac{y^{2}}{16} - \frac{z^{2}}{25} \]The standard form.✓ Proved
- A hyperboloid of one sheet: two squares positive and one negative, = 1.Reviewed
Answer \( \frac{x^{2}}{9} + \frac{y^{2}}{16} - \frac{z^{2}}{25} = 1\ (\text{hyperboloid of one sheet}) \)
✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the surface classified from the signs of its coefficients; the standard form is a multiple of the given equation |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly divides by the constant term to obtain the standard form and accurately identifies the surface as a hyperboloid of one sheet based on the signs of the coefficients.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly divides by the constant term to obtain the standard form and accurately identifies the surface as a hyperboloid of one sheet based on the signs of the coefficients.gpt-oss:20b: pass 2026-11-06qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly derives the standard form and accurately identifies the surface based on the signs of the coefficients.gpt-oss:20b: pass 2026-11-05qwen3.6:27b-mlx: pass 2026-11-05 — The algebraic manipulation is correct, and the identification of the surface as a hyperboloid of one sheet is accurate based on the standard form.gpt-oss:20b: pass 2026-11-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/quadric_surfaces, checked 2026-11-06 with SymPy 1.14.0.