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Equations of spheres

Problem 9.919 · easy

Find the equation of the sphere with center \( \displaystyle (4, 2, 0) \) 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.
Reviewed
\[ \left. z^{2} + \left(x - 4\right)^{2} + \left(y - 2\right)^{2} \right|_{\substack{ x=5\\ y=2\\ z=0 }} = 1 \]
A point at distance r along the x-direction is on it.✓ Proved

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

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 verification step confirms a point on the sphere satisfies the equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius. The verification step confirms a point on the sphere satisfies the equation.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly identifies the standard equation of a sphere and verifies it with a valid point on the sphere. The logic is sound and the final answer matches the problem statement.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius. The unchecked equation line serves as a valid consistency check for a point on the sphere.
gpt-oss:20b: fail (error) 2026-11-07 — The solution never writes the sphere’s equation; it merely attempts a substitution without showing the result, and the justification that (5,2,0) lies on the sphere is missing.

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 2026-11-08 with SymPy 1.14.0.