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Level curves and domains

Problem 10.630 · easy Video

Find the equation of the level curve of \( \displaystyle f(x, y) = - x^{2} - 4 y^{2} + 11 \) that passes through \( \displaystyle (2, 1) \), and describe it.

A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. - x^{2} - 4 y^{2} + 11 \right|_{\substack{ x=2\\ y=1 }} = 3 \]
c = f(2, 1).✓ Proved
So the level curve is Eq(-x**2 - 4*y**2 + 11, 3), a ellipse.
Reviewed

Answer \( - x^{2} - 4 y^{2} + 11 = 3\ (\text{a ellipse}) \)

A contour map of f(x, y) = -x² - 4y² + 11: each curve joins points where f has the same value. The level curve through (2, 1) is highlighted.

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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the constant c by evaluating the function at the given point and correctly identifies the resulting 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 equation as an ellipse.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to simplify the equation into standard form, leaving it as an implicit relation rather than the explicit equation of the curve. Additionally, 'a ellipse' is grammatically incorrect; it should be 'an ellipse'.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-14 — The article 'a' is incorrect before 'ellipse'; it should be 'an ellipse'. While mathematically sound, this grammatical error is a style issue.
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 2026-10-18 with SymPy 1.14.0.