Quadric surfaces in standard form
Problem 9.729 · easy
Write \( \displaystyle 75 x^{2} + 12 y^{2} - 75 z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 300 and solve for the variable to the first power.
- \[ \frac{75 x^{2} + 12 y^{2} - 75 z^{2}}{300} = \frac{x^{2}}{4} + \frac{y^{2}}{25} - \frac{z^{2}}{4} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( \frac{x^{2}}{4} + \frac{y^{2}}{25} - \frac{z^{2}}{4} = 0\ (\text{elliptic cone}) \)
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 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), so this instruction is nonsensical and misrepresents the algebraic operation performed.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — 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), so this instruction is nonsensical and misrepresents the algebraic operation performed.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The instruction in step 1 to 'solve for the variable to the first power' is incorrect because there are no first-power terms in the equation; the equation is homogeneous. Additionally, step 3 contains a grammatical error ('A elliptic cone').gpt-oss:20b: fail (error) 2026-10-16 — Sentence 1 incorrectly states “solve for the variable to the first power,” which is unnecessary and misleading; the step should simply be dividing both sides by 300 to obtain the standard form. The rest of the solution is correct.qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — The instruction to 'solve for the variable to the first power' is incorrect because there are no first-power terms in the equation; the standard form is obtained simply by dividing by the constant term on the RHS (0) to normalize the coefficients, not by isolating a linear variable.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.