Quadric surfaces in standard form
Problem 9.649 · easy
Write \( \displaystyle 25 x^{2} + 36 y^{2} - 100 z^{2} = 0 \) in standard form and identify the surface.
- Divide through by 900 and solve for the variable to the first power.
- \[ \frac{25 x^{2} + 36 y^{2} - 100 z^{2}}{900} = \frac{x^{2}}{36} + \frac{y^{2}}{25} - \frac{z^{2}}{9} \]The standard form.✓ Proved
- A elliptic cone: two squares minus one, = 0.
Answer \( \frac{x^{2}}{36} + \frac{y^{2}}{25} - \frac{z^{2}}{9} = 0\ (\text{elliptic cone}) \)
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 claims to 'solve for the variable to the first power', but no variable appears to the first power in the equation, and the equation is not solved for any variable. The description of the cone in step 3 ('two squares minus one') is imprecise and potentially confusing.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The sentence in step 1 claims to 'solve for the variable to the first power', but no variable appears to the first power in the equation, and the equation is not solved for any variable. The description of the cone in step 3 ('two squares minus one') is imprecise and potentially confusing.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The instruction in step 1 to 'solve for the variable to the first power' is incorrect; the equation is homogeneous of degree 2, so no variable appears to the first power, and the standard form is obtained by simple division, not solving for a variable.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The instruction to 'solve for the variable to the first power' is incorrect because there are no first-power terms in the equation; the equation is homogeneous of degree 2. Additionally, the description 'two squares minus one' is imprecise and potentially misleading regarding the coefficients.gpt-oss:20b: fail (misleading) 2026-10-14 — The first sentence claims to "solve for the variable to the first power," which is unnecessary and could confuse a student; the rest of the solution is correct but the wording is misleading.
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.