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Quadric surfaces in standard form

Problem 9.831 · easy

Write \( \displaystyle 75 x^{2} + 108 y^{2} - 108 z^{2} = 0 \) in standard form and identify the surface.

Divide through by 2700 and solve for the variable to the first power.
Reviewed
\[ \frac{75 x^{2} + 108 y^{2} - 108 z^{2}}{2700} = \frac{x^{2}}{36} + \frac{y^{2}}{25} - \frac{z^{2}}{25} \]
The standard form.✓ Proved
A elliptic cone: two squares minus one, = 0.
Reviewed

Answer \( \frac{x^{2}}{36} + \frac{y^{2}}{25} - \frac{z^{2}}{25} = 0\ (\text{elliptic cone}) \)

Lines: 1 proved, 2 reviewed. 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	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: pass
qwen3.6:27b-mlx: pass — The solution correctly transforms the equation into standard form and accurately identifies the surface as an elliptic cone.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly transforms the equation into standard form and accurately identifies the surface as an elliptic cone.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (style) 2026-10-19 — The instruction in step 1 ('solve for the variable to the first power') is incorrect because there are no first-power variables in the equation; the goal is simply to normalize the coefficients to 1. Step 3's description of the form is imprecise.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The sentence in step 1 claims to 'solve for the variable to the first power', but the equation contains no linear terms (all variables are squared). This is a nonsensical instruction for converting a quadratic form to standard form.
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.