Quadric surfaces in standard form
Problem 9.728 · easy
Write \( \displaystyle 2 x^{2} + 2 y^{2} - 8 z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 32 and solve for the variable to the first power.
- \[ \frac{2 x^{2} + 2 y^{2} - 8 z^{2}}{32} = \frac{x^{2}}{16} + \frac{y^{2}}{16} - \frac{z^{2}}{4} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( \frac{x^{2}}{16} + \frac{y^{2}}{16} - \frac{z^{2}}{4} = 0\ (\text{elliptic cone}) \)
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 instruction in step 1 ('solve for the variable to the first power') is incorrect because all variables are squared; one simply divides by the constant term to normalize the equation. Additionally, step 3's heuristic 'two squares minus one' is imprecise and potentially misleading regarding the coefficients.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The instruction in step 1 ('solve for the variable to the first power') is incorrect because all variables are squared; one simply divides by the constant term to normalize the equation. Additionally, step 3's heuristic 'two squares minus one' is imprecise and potentially misleading regarding the coefficients.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly transforms the equation into standard form and accurately identifies the surface as an elliptic cone.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly divides by 32 instead of 8 to obtain the standard form. Dividing by 32 yields coefficients of 1/16 and -1/4, whereas dividing by 8 yields coefficients of 1/4 and -1. While the surface identification is correct, the derived equation is not the standard form requested.gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence claims to "solve for the variable to the first power," but the step only divides the equation by 32; it does not solve for a variable. The wording is misleading and could confuse a student.
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.