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Level curves and domains

Problem 10.711 · easy Video

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

A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. 2 x^{2} + y^{2} \right|_{\substack{ x=1\\ y=2 }} = 6 \]
c = f(1, 2).✓ Proved
So the level curve is Eq(2*x**2 + y**2, 6), a ellipse.
Reviewed

Answer \( 2 x^{2} + y^{2} = 6\ (\text{a ellipse}) \)

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

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 level curve equation by evaluating the function at the given point and accurately describes the resulting conic section as an ellipse.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the level curve equation by evaluating the function at the given point and accurately describes the resulting conic section as an ellipse.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-17 — The article 'a' should be 'an' before 'ellipse'. While mathematically correct, the grammatical error is a style issue.
gpt-oss:20b: pass 2026-10-17
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 equation as an ellipse.
gpt-oss:20b: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.