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Equations of spheres

Problem 9.804 · easy

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

A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
\[ \left. \left(x + 1\right)^{2} + \left(y + 4\right)^{2} + \left(z - 2\right)^{2} \right|_{\substack{ x=0\\ y=-4\\ z=2 }} = 1 \]
A point at distance r along the x-direction is on it.✓ Proved

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

Lines: 1 proved, 1 not checked. 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	Not checked	—	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: fail (error) — The solution fails to substitute the given center (-1, -4, 2) into the standard equation. Instead, it verifies a point (0, -4, 2) which is not the center, and the equation line is syntactically malformed as a verification step rather than the derivation of the final answer.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to substitute the given center (-1, -4, 2) into the standard equation. Instead, it verifies a point (0, -4, 2) which is not the center, and the equation line is syntactically malformed as a verification step rather than the derivation of the final answer.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius. The verification step in line 2 is valid, as the point (0, -4, 2) is indeed at distance 1 from the center (-1, -4, 2).
gpt-oss:20b: fail (error) 2026-10-16 — The solution never derives the sphere’s equation from the given center and radius; it only verifies a single point. The equation itself is missing from the logical steps, so the setup is incomplete.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution attempts to verify the equation by substituting a point (0, -4, 2) into the left-hand side, but the syntax is malformed and the logic is flawed. The point (0, -4, 2) is indeed on the sphere (distance 1 from center), but the 'equation' line is not a standard mathematical statement and the substitution check is presented as a derivation step rather than a verification, which is confusing. More importantly, the problem asks for the equation, which is derived directly from the definition in step 1; step 2 is an unnecessary and poorly formatted verification that adds no value and introduces syntax errors.
gpt-oss:20b: pass 2026-10-15

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 2026-10-19 with SymPy 1.14.0.