Level curves and domains
Problem 10.667 · easy Video
Find the equation of the level curve of \( \displaystyle f(x, y) = - 3 x^{2} - 4 y^{2} + 5 \) that passes through \( \displaystyle (1, 2) \), and describe it.
- A level curve is f(x, y) = c; the point fixes c.Reviewed
- \[ \left. - 3 x^{2} - 4 y^{2} + 5 \right|_{\substack{ x=1\\ y=2 }} = -14 \]c = f(1, 2).✓ Proved
- So the level curve is Eq(-3*x**2 - 4*y**2 + 5, -14), a ellipse.Reviewed
Answer \( - 3 x^{2} - 4 y^{2} + 5 = -14\ (\text{a ellipse}) \)
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: fail (error) 2026-10-16 — The equation -3x^2 - 4y^2 + 5 = -14 simplifies to 3x^2 + 4y^2 = 19, which is an ellipse. However, the solution presents the unsimplified equation as the final answer and incorrectly labels the unsimplified form '-3x^2 - 4y^2 + 5 = -14' as 'a ellipse' without showing the standard form derivation, which is misleading. More critically, the prompt asks for 'the equation', and while the implicit form is correct, the description 'a ellipse' is grammatically poor ('an ellipse') and the lack of simplification makes it a poor model for a student. But the main error is likely that the user might expect the standard form. Let's look closer. Is it an error? No, it's a valid equation. Is it misleading? Yes, because it doesn't show the standard form. Is it an error? The classification 'ellipse' is correct. The grammar 'a ellipse' is a style issue. Wait, let's re-read the prompt constraints. 'Find the equation... and describe it.' The equation given is correct. The description is correct. The grammar is bad. Is there a deeper error? No. But usually, 'find the equation' implies simplifying. Let's check if the level curve is actually an ellipse. 3x^2 + 4y^2 = 19. Yes. So the math is correct. The only issue is style/grammar. However, often in these adversarial reviews, if the answer isn't simplified, it might be considered 'misleading' or 'style'. Let's look at the options. 'error' is for wrong steps. 'misleading' is for false learning. 'style' is for clarity. The sentence 'So the level curve is Eq(...), a ellipse' is grammatically incorrect and doesn't simplify. I will mark it as style because the math is fundamentally correct, just poorly presented.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — 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 2026-10-15
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.