Quadric surfaces in standard form
Problem 9.726 · easy
Write \( \displaystyle 2 x^{2} + 2 y^{2} - 2 z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 72 and solve for the variable to the first power.
- \[ \frac{2 x^{2} + 2 y^{2} - 2 z^{2}}{72} = \frac{x^{2}}{36} + \frac{y^{2}}{36} - \frac{z^{2}}{36} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( \frac{x^{2}}{36} + \frac{y^{2}}{36} - \frac{z^{2}}{36} = 0\ (\text{elliptic cone}) \)
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: fail (misleading) — The first sentence incorrectly states that one must divide by 72 and “solve for the variable to the first power.” The equation can be divided by 2 (or 72) to obtain the standard form, but no solving for a first‑power variable is required. The wording may mislead a student into thinking a different manipulation is necessary.qwen3.6:27b-mlx: fail (error) — The instruction to 'solve for the variable to the first power' is incorrect because the equation is homogeneous of degree 2; no variable appears to the first power. Additionally, dividing by 72 is arbitrary and unnecessary to reach standard form (dividing by 2 suffices).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The instruction to 'solve for the variable to the first power' is incorrect because the equation is homogeneous of degree 2; no variable appears to the first power. Additionally, dividing by 72 is arbitrary and unnecessary to reach standard form (dividing by 2 suffices).gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence incorrectly states that one must divide by 72 and “solve for the variable to the first power.” The equation can be divided by 2 (or 72) to obtain the standard form, but no solving for a first‑power variable is required. The wording may mislead a student into thinking a different manipulation is necessary.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The instruction to 'solve for the variable to the first power' is incorrect because there are no first-power terms in the equation. Additionally, dividing by 72 is arbitrary; dividing by 2 is the standard step to reach the simplest standard form.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The instruction to 'solve for the variable to the first power' is incorrect because the equation is homogeneous (all terms are degree 2); no variable appears to the first power. Additionally, dividing by 72 is arbitrary and unnecessary for standard form, which typically requires the RHS to be 1 (for non-degenerate quadrics) or simply simplifying coefficients.gpt-oss:20b: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly divides by 72 instead of 2, leading to a non‑standard form. The standard form should be \(x^{2}+y^{2}-z^{2}=0\) (or with denomina
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.