Quadric surfaces in standard form
Problem 9.652 · easy
Write \( \displaystyle x^{2} + 4 y^{2} - z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 36 and solve for the variable to the first power.
- \[ \frac{x^{2} + 4 y^{2} - z^{2}}{36} = \frac{x^{2}}{36} + \frac{y^{2}}{9} - \frac{z^{2}}{36} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( \frac{x^{2}}{36} + \frac{y^{2}}{9} - \frac{z^{2}}{36} = 0\ (\text{elliptic cone}) \)
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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly divides by 36 without justification, altering the equation's coefficients arbitrarily. The original equation is already in a standard form for a cone (or can be written as z^2 = x^2 + 4y^2), but the derived equation is not equivalent to the original unless specific scaling was requested, which it was not.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly divides by 36 without justification, altering the equation's coefficients arbitrarily. The original equation is already in a standard form for a cone (or can be written as z^2 = x^2 + 4y^2), but the derived equation is not equivalent to the original unless specific scaling was requested, which it was not.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution arbitrarily divides by 36 without justification from the problem statement, producing a specific instance rather than the general standard form. Additionally, the identification rule 'two squares minus one' is imprecise and ignores the coefficients required for an elliptic cone.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution arbitrarily divides by 36 without justification, as the original equation does not contain this factor. The standard form of x^2 + 4y^2 - z^2 = 0 is simply itself (or x^2/1 + y^2/(1/4) - z^2/1 = 0), not the version with 36 in the denominators.gpt-oss:20b: fail (error) 2026-10-14 — The standard form of a cone should be x²/a² + y²/b² = z²/c² (or equivalently x²/a² + y²/b² – z²/c² = 0). Dividing by 36 does not change the equation but does not put it into standard form; the given form is simply a scaled version of the original. The sentence about “solving for the variable to the first power” is unnecessary and the description of the surface is vague.
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 2026-10-17 with SymPy 1.14.0.