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Equations of spheres

Problem 9.805 · easy

Find the equation of the sphere with center \( \displaystyle (1, -4, 5) \) and radius \( \displaystyle 3 \), in standard form.

A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
\[ \left. \left(x - 1\right)^{2} + \left(y + 4\right)^{2} + \left(z - 5\right)^{2} \right|_{\substack{ x=4\\ y=-4\\ z=5 }} = 9 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer \( \left(x - 1\right)^{2} + \left(y + 4\right)^{2} + \left(z - 5\right)^{2} = 9 \)

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 explicitly substitute the given center (1, -4, 5) and radius 3 into the standard equation formula. Instead, it verifies a single point (4, -4, 5) which happens to be on the sphere, but this verification does not constitute deriving the equation from the problem statement.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to explicitly substitute the given center (1, -4, 5) and radius 3 into the standard equation formula. Instead, it verifies a single point (4, -4, 5) which happens to be on the sphere, but this verification does not constitute deriving the equation from the problem statement.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (misleading) 2026-10-16 — The solution provides the correct final answer but includes a confusing and logically irrelevant step (checking a specific point) that does not constitute a derivation of the equation. It fails to explicitly show the substitution of the given center and radius into the standard formula, which is the core of the problem.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — 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.
gpt-oss:20b: fail (misleading) 2026-10-15 — The sentence claiming a point at distance r along the x‑direction lies on the sphere is not justified; the solution merely substitutes a point without verifying its distance from the center.

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.