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Equations of spheres

Problem 9.625 · easy

Find the equation of the sphere with center \( \displaystyle (3, -4, 1) \) and radius \( \displaystyle 2 \), in standard form.

A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
Reviewed
\[ \left. \left(x - 3\right)^{2} + \left(y + 4\right)^{2} + \left(z - 1\right)^{2} \right|_{\substack{ x=5\\ y=-4\\ z=1 }} = 4 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer \( \left(x - 3\right)^{2} + \left(y + 4\right)^{2} + \left(z - 1\right)^{2} = 4 \)

Lines: 1 proved, 1 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	points at the claimed radius from the claimed center satisfy the equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius. The algebraic verification in line 2 is consistent with the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius. The algebraic verification in line 2 is consistent with the stated answer.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly state the final equation derived from the standard form formula. It only provides a verification step for a single point, which does not constitute finding the equation as requested.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius. The unchecked equation serves as a valid verification step for a specific point on the sphere.
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/sphere_equation, checked 2026-10-17 with SymPy 1.14.0.