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Level curves and domains

Problem 10.637 · easy Video

Find the equation of the level curve of \( \displaystyle f(x, y) = - x^{2} + y^{2} \) that passes through \( \displaystyle (-1, 3) \), and describe it.

A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. - x^{2} + y^{2} \right|_{\substack{ x=-1\\ y=3 }} = 8 \]
c = f(-1, 3).✓ Proved
So the level curve is Eq(-x**2 + y**2, 8), a hyperbola.
Reviewed

Answer \( - x^{2} + y^{2} = 8\ (\text{a hyperbola}) \)

A contour map of f(x, y) = -x² + y²: each curve joins points where f has the same value. The level curve through (-1, 3) 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 correctly classifies the resulting conic section as a hyperbola.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the level curve equation by evaluating the function at the given point and correctly classifies the resulting conic section as a hyperbola.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the constant c by evaluating the function at the given point and correctly identifies the resulting equation as a hyperbola.
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.