Quadric surfaces in standard form
Problem 9.653 · easy
Write \( \displaystyle 18 x^{2} + 288 y^{2} - 32 z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 288 and solve for the variable to the first power.
- \[ \frac{18 x^{2} + 288 y^{2} - 32 z^{2}}{288} = \frac{x^{2}}{16} + y^{2} - \frac{z^{2}}{9} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( \frac{x^{2}}{16} + y^{2} - \frac{z^{2}}{9} = 0\ (\text{elliptic cone}) \)
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: passqwen3.6:27b-mlx: fail (error) — The standard form of an elliptic cone is typically written with positive coefficients for the spatial variables equal to 1 (e.g., x^2/a^2 + y^2/b^2 = z^2/c^2), not equal to 0 with mixed signs, although the latter is algebraically equivalent. More critically, the sentence 'two squares minus one, = 0' is confusingly phrased and potentially misleading regarding the structure of the equation (it implies a constant term of -1, whereas the equation is homogeneous).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The standard form of an elliptic cone is typically written with positive coefficients for the spatial variables equal to 1 (e.g., x^2/a^2 + y^2/b^2 = z^2/c^2), not equal to 0 with mixed signs, although the latter is algebraically equivalent. More critically, the sentence 'two squares minus one, = 0' is confusingly phrased and potentially misleading regarding the structure of the equation (it implies a constant term of -1, whereas the equation is homogeneous).gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The setup instruction in step 1 is incorrect; one does not 'solve for the variable to the first power' to obtain the standard form of a quadric surface, one simply divides by the constant term to normalize the equation. Additionally, step 3 contains a grammatical error ('A elliptic') and an imprecise description of the signature.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The standard form of an elliptic cone is typically written with positive coefficients for the squared terms on one side and the negative term on the other (e.g., x^2/a^2 + y^2/b^2 = z^2/c^2), or explicitly as a sum equal to zero with correct signs. While the equation derived is algebraically equivalent, the description 'two squares minus one' is imprecise and potentially confusing regarding the standard form structure. More importantly, the prompt asks to write it in standard form; the provided equation `x^2/16 + y^2 - z^2/9 = 0` is acceptable, but the sentence in step 1 says 'solve for the variable to the first power', which is nonsensical as there are no first-power variables. The instruction is flawed.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.