Quadric surfaces in standard form
Problem 9.907 · easy
Write \( \displaystyle 8 x^{2} + 18 y^{2} - 18 z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 72 and solve for the variable to the first power.
- \[ \frac{8 x^{2} + 18 y^{2} - 18 z^{2}}{72} = \frac{x^{2}}{9} + \frac{y^{2}}{4} - \frac{z^{2}}{4} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( \frac{x^{2}}{9} + \frac{y^{2}}{4} - \frac{z^{2}}{4} = 0\ (\text{elliptic cone}) \)
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 to 'solve for the variable to the first power' is incorrect because there are no first-power variables in the equation; the standard form is achieved simply by dividing by the constant term to make the RHS 1 (or 0). Step 3's description 'two squares minus one' is also imprecise phrasing for the signature of the quadratic form.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-06 — The instruction in step 1 to 'solve for the variable to the first power' is incorrect because there are no first-power variables in the equation; the standard form is achieved simply by dividing by the constant term to make the RHS 1 (or 0). Step 3's description 'two squares minus one' is also imprecise phrasing for the signature of the quadratic form.gpt-oss:20b: pass 2026-11-06qwen3.6:27b-mlx: fail (error) 2026-11-05 — 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. The correct procedure is simply to divide by the constant term to normalize the equation to equal 1 (or 0), not to solve for a variable.gpt-oss:20b: pass 2026-11-05qwen3.6:27b-mlx: fail (error) 2026-11-05 — 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). This instruction is nonsensical for this problem.gpt-oss:20b: fail (misleading) 2026-11-05 — The first sentence claims to "solve for the variable to the first power," which is incorrect; dividing by 72 only rewrites the equation, it does not solve for a variable. The rest of the solution is essentially correct, but the misleading statement 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 2026-11-06 with SymPy 1.14.0.