Quadric surfaces in standard form
Problem 9.733 · easy
Write \( \displaystyle 400 x^{2} + 100 y^{2} - 16 z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 400 and solve for the variable to the first power.
- \[ \frac{400 x^{2} + 100 y^{2} - 16 z^{2}}{400} = x^{2} + \frac{y^{2}}{4} - \frac{z^{2}}{25} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( x^{2} + \frac{y^{2}}{4} - \frac{z^{2}}{25} = 0\ (\text{elliptic cone}) \)
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 instructs 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 does not describe the algebraic operation performed. Additionally, step 3 contains a grammatical error ('A elliptic cone').
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The sentence in step 1 instructs 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 does not describe the algebraic operation performed. Additionally, step 3 contains a grammatical error ('A elliptic cone').gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (misleading) 2026-10-16 — The instruction in step 1 to 'solve for the variable to the first power' is nonsensical because there are no first-power terms in the equation. Step 3's heuristic 'two squares minus one' is imprecise and ignores the coefficients, which determine whether it is an elliptic or circular cone.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — 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). The standard form for a cone is typically written as z^2 = c(x^2/a^2 + y^2/b^2) or similar, not just by dividing coefficients.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.