Level curves and domains
Problem 10.662 · easy Video
Find the equation of the level curve of \( \displaystyle f(x, y) = - 2 x^{2} - 3 y^{2} + 5 \) that passes through \( \displaystyle (2, -1) \), and describe it.
- A level curve is f(x, y) = c; the point fixes c.Reviewed
- \[ \left. - 2 x^{2} - 3 y^{2} + 5 \right|_{\substack{ x=2\\ y=-1 }} = -6 \]c = f(2, -1).✓ Proved
- So the level curve is Eq(-2*x**2 - 3*y**2 + 5, -6), a ellipse.Reviewed
Answer \( - 2 x^{2} - 3 y^{2} + 5 = -6\ (\text{a ellipse}) \)
Lines: 1 proved, 2 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | points found numerically on the claimed curve near P give f the same value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the constant c by evaluating the function at the given point and correctly identifies the resulting quadratic equation as an ellipse.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the constant c by evaluating the function at the given point and correctly identifies the resulting quadratic equation as an ellipse.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the constant c by evaluating the function at the given point and correctly classifies the resulting quadratic equation as an ellipse.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The equation -2x^2 - 3y^2 + 5 = -6 simplifies to 2x^2 + 3y^2 = 11, which is an ellipse. However, the solution presents the equation in a non-standard form and incorrectly labels it 'a ellipse' (grammar aside, the description is technically correct but the phrasing is poor). More importantly, the prompt asks for 'the equation', and while the implicit form is given, standard practice usually requires simplifying to standard form (e.g., x^2/a^2 + y^2/b^2 = 1) or at least moving constants. But the main issue is likely the grammar 'a ellipse' vs 'an ellipse' or the lack of simplification. Let's look closer. Is it an error? No, the curve is an ellipse. Is it misleading? No. Is it style? Yes, 'a ellipse' is bad grammar, and leaving the equation as -2x^2 - 3y^2 + 5 = -6 is not simplified. However, usually 'error' is reserved for mathematical mistakes. Let's re-read the prompt constraints. 'Find what is wrong'. The math is correct. The description 'a ellipse' is grammatically incorrect. The equation is not simplified. Is this an error? In many contexts, leaving an equation unsimplified is considered a style issue or a minor error. But wait, looking at the provided solution line 3: 'So the level curve is Eq(-2*x**2 - 3*y**2 + 5, -6), a ellipse.' The math is sound. The grammar is poor. I will mark it as style.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/level_curves, checked 2026-10-18 with SymPy 1.14.0.