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Quadric surfaces in standard form

Problem 9.829 · easy

Write \( \displaystyle 2 x^{2} + 72 y^{2} - 8 z^{2} = 0 \) in standard form and identify the surface.

Divide through by 72 and solve for the variable to the first power.
\[ \frac{2 x^{2} + 72 y^{2} - 8 z^{2}}{72} = \frac{x^{2}}{36} + y^{2} - \frac{z^{2}}{9} \]
The standard form.✓ Proved
A elliptic cone: two squares minus one, = 0.

Answer \( \frac{x^{2}}{36} + y^{2} - \frac{z^{2}}{9} = 0\ (\text{elliptic cone}) \)

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: pass
qwen3.6:27b-mlx: fail (error) — The standard form of a quadratic surface equal to 0 should typically be written with positive 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 of squares equal to zero with clear coefficients. While the algebraic manipulation in step 2 is correct, step 3's description 'two squares minus one, = 0' is ambiguous and potentially misleading regarding the standard form convention. More importantly, the prompt asks to 'write in standard form', and the provided equation x^2/36 + y^2 - z^2/9 = 0 is technically a valid standard form for a cone, but the explanation in step 3 is poor. However, the primary issue is that step 1 says 'solve for the variable to the first power', but no variable is to the first power in the original equation (all are squared). This is a factual error in the setup description.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-20 — The standard form of a quadratic surface equal to 0 should typically be written with positive 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 of squares equal to zero with clear coefficients. While the algebraic manipulation in step 2 is correct, step 3's description 'two squares minus one, = 0' is ambiguous and potentially misleading regarding the standard form convention. More importantly, the prompt asks to 'write in standard form', and the provided equation x^2/36 + y^2 - z^2/9 = 0 is technically a valid standard form for a cone, but the explanation in step 3 is poor. However, the primary issue is that step 1 says 'solve for the variable to the first power', but no variable is to the first power in the original equation (all are squared). This is a factual error in the setup description.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The standard form of a cone is typically written as z^2 = c(x^2/a^2 + y^2/b^2) or similar, isolating the squared term on one side, rather than setting the sum to zero. More importantly, the description 'two squares minus one' is mathematically imprecise and confusing; it should describe the signs of the coefficients (two positive, one negative) in the equation equal to zero.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The sentence in step 1 is mathematically incoherent; one does not 'solve for the variable to the first power' when dividing by a constant to reach standard form. Additionally, step 3 misidentifies the signature of the cone (it has two positive terms and one negative, not 'two squares minus one' in the sense of signature 2,1 vs 1,2, though the final classification is correct, the reasoning is garbled).
gpt-oss:20b: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.