Quadric surfaces in standard form
Problem 9.906 · easy
Write \( \displaystyle 108 x^{2} + 27 y^{2} - 12 z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 108 and solve for the variable to the first power.
- \[ \frac{108 x^{2} + 27 y^{2} - 12 z^{2}}{108} = x^{2} + \frac{y^{2}}{4} - \frac{z^{2}}{9} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( x^{2} + \frac{y^{2}}{4} - \frac{z^{2}}{9} = 0\ (\text{elliptic cone}) \)
Lines: 1 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | 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: fail (misleading) — The first sentence incorrectly states “solve for the variable to the first power,” which is unnecessary and misleading. The rest of the solution correctly identifies the surface as an elliptic cone.qwen3.6:27b-mlx: pass — The solution correctly derives the standard form and accurately identifies the surface as an elliptic cone based on the signature of the quadratic terms.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly derives the standard form and accurately identifies the surface as an elliptic cone based on the signature of the quadratic terms.gpt-oss:20b: fail (misleading) 2026-11-06 — The first sentence incorrectly states “solve for the variable to the first power,” which is unnecessary and misleading. The rest of the solution correctly identifies the surface as an elliptic cone.qwen3.6:27b-mlx: fail (error) 2026-11-05 — The solution fails to actually solve for a variable as stated in step 1; it merely divides by 108. While the resulting equation is a valid standard form, the description 'two squares minus one' is imprecise and potentially confusing (it should specify the signs, e.g., 'sum of two squares equals a third square').gpt-oss:20b: pass 2026-11-05qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly derives the standard form and accurately identifies the surface as an elliptic cone based on the signs of the squared terms.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.