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Equations of spheres

Problem 9.845 · easy

Find the equation of the sphere with center \( \displaystyle (4, -5, -1) \) and radius \( \displaystyle 5 \), 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
\[ \left. \left(x - 4\right)^{2} + \left(y + 5\right)^{2} + \left(z + 1\right)^{2} \right|_{\substack{ x=9\\ y=-5\\ z=-1 }} = 25 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer \( \left(x - 4\right)^{2} + \left(y + 5\right)^{2} + \left(z + 1\right)^{2} = 25 \)

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 to derive the correct final answer. The verification step in line 2 is valid.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius to derive the correct final answer. The verification step in line 2 is valid.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution provides the correct final equation but fails to explicitly state the substitution of the center coordinates (h, k, l) = (4, -5, -1) and radius r = 5 into the standard formula. Instead, it verifies a single point, which is not a derivation of the equation. A student might incorrectly infer that checking one point is sufficient to define the sphere's equation.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to explicitly substitute the given center coordinates and radius into the standard formula to derive the final equation. Instead, it provides an unchecked verification step for a single point, which does not constitute a derivation of the equation.
gpt-oss:20b: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.