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Equations of spheres

Problem 9.622 · easy

Find the equation of the sphere with center \( \displaystyle (0, 4, 0) \) 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.
\[ \left. x^{2} + z^{2} + \left(y - 4\right)^{2} \right|_{\substack{ x=5\\ y=4\\ z=0 }} = 25 \]
A point at distance r along the x-direction is on it.✓ Proved

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

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 derive the equation of the sphere. It states the general formula but then provides an irrelevant equation checking a single point, rather than substituting the center (0, 4, 0) and radius 5 into the standard form to produce the final answer.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to derive the equation of the sphere. It states the general formula but then provides an irrelevant equation checking a single point, rather than substituting the center (0, 4, 0) and radius 5 into the standard form to produce the final answer.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to derive the equation of the sphere. Step 2 checks a single point on the sphere rather than substituting the center and radius into the general formula to obtain the equation.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly substitutes the radius (5) for the x-coordinate and the center's y-coordinate (4) for the y-coordinate in a point-check, rather than substituting the center coordinates into the equation to verify the form. This demonstrates a fundamental misunderstanding of how to validate the equation of a 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.