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Quadric surfaces in standard form

Problem 9.904 · easy

Write \( \displaystyle 81 x^{2} + 9 y^{2} - 9 z^{2} = 0 \) in standard form and identify the surface.

Divide through by 81 and solve for the variable to the first power.
\[ \frac{81 x^{2} + 9 y^{2} - 9 z^{2}}{81} = x^{2} + \frac{y^{2}}{9} - \frac{z^{2}}{9} \]
The standard form.✓ Proved
A elliptic cone: two squares minus one, = 0.

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

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 (misleading) — The solution fails to explicitly write the standard form equation equal to 1 (or 0 in the specific cone form), but more importantly, the identification logic in step 3 is imprecise. While the resulting equation is correct for an elliptic cone, the description 'two squares minus one, = 0' is vague and does not strictly define the standard form structure (e.g., x^2/a^2 + y^2/b^2 - z^2/c^2 = 0) required to distinguish it from other quadric surfaces. A student might confuse this with a hyperboloid if the '=0' condition isn't emphasized as the defining feature of a cone versus a hyperboloid of one sheet.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-11-06 — The solution fails to explicitly write the standard form equation equal to 1 (or 0 in the specific cone form), but more importantly, the identification logic in step 3 is imprecise. While the resulting equation is correct for an elliptic cone, the description 'two squares minus one, = 0' is vague and does not strictly define the standard form structure (e.g., x^2/a^2 + y^2/b^2 - z^2/c^2 = 0) required to distinguish it from other quadric surfaces. A student might confuse this with a hyperboloid if the '=0' condition isn't emphasized as the defining feature of a cone versus a hyperboloid of one sheet.
gpt-oss:20b: pass 2026-11-06
qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly derives the standard form and accurately identifies the surface as an elliptic cone based on the signature of the quadratic terms.
gpt-oss:20b: pass 2026-11-05
qwen3.6:27b-mlx: fail (error) 2026-11-05 — The instruction to 'solve for the variable to the first power' is incorrect because all variables are squared; the equation represents a cone, not a function solvable for a linear variable. Additionally, the classification rule 'two squares minus one' is imprecise and potentially misleading regarding the signs.
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.