Level curves and domains
Problem 10.570 · easy Video
Find the equation of the level curve of \( \displaystyle f(x, y) = - 4 x^{2} - 2 y^{2} + 10 \) that passes through \( \displaystyle (1, 2) \), and describe it.
- A level curve is f(x, y) = c; the point fixes c.Reviewed
- \[ \left. - 4 x^{2} - 2 y^{2} + 10 \right|_{\substack{ x=1\\ y=2 }} = -2 \]c = f(1, 2).✓ Proved
- So the level curve is Eq(-4*x**2 - 2*y**2 + 10, -2), a ellipse.Reviewed
Answer \( - 4 x^{2} - 2 y^{2} + 10 = -2\ (\text{a ellipse}) \)
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 2026-10-17 — 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 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the constant c by evaluating the function at the given point and states the resulting equation. The classification of the curve as an ellipse is correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The description 'a ellipse' is grammatically incorrect and mathematically imprecise; it should be 'an ellipse'. More importantly, the solution fails to explicitly state the standard form of the ellipse (e.g., 2x^2 + y^2 = 6) which is typically required to 'describe' the curve fully (center, axes, etc.), though the equation itself is correct. However, the primary failure is the grammatical error 'a ellipse' which is a style issue, but often in these strict reviews, 'describe it' implies identifying the shape correctly. Let's look closer. Is 'a ellipse' an error or style? It's a style/grammar error. But is the math wrong? No. Is the description sufficient? 'An ellipse' is the shape. The prompt asks to 'describe it'. Saying it is an ellipse is a valid description. The grammar 'a ellipse' is poor. Let's check if there are deeper issues. The equation -4x^2 - 2y^2 + 10 = -2 simplifies to 4x^2 + 2y^2 = 12 or 2x^2 + y^2 = 6. This is indeed an ellipse. The solution provides the equation and the shape. The only flaw is 'a ellipse'. In many automated grading contexts, 'a ellipse' might be flagged as a style error. However, looking at the severity options: 'error' is for wrong steps/setup. 'misleading' is for false teaching. 'style' is for clarity. 'a ellipse' is a style issue. But wait, is the verdict 'pass' or 'fail'? The instructions say 'Say pass with empty notes if it is sound.' Is 'a ellipse' sound? It's grammatically unsound. I will mark it as fail/style.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.