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Equations of spheres

Problem 9.846 · easy

Find the equation of the sphere with center \( \displaystyle (2, -2, -5) \) and radius \( \displaystyle 4 \), in standard form.

A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
\[ \left. \left(x - 2\right)^{2} + \left(y + 2\right)^{2} + \left(z + 5\right)^{2} \right|_{\substack{ x=6\\ y=-2\\ z=-5 }} = 16 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer \( \left(x - 2\right)^{2} + \left(y + 2\right)^{2} + \left(z + 5\right)^{2} = 16 \)

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 (misleading) — The solution provides the correct final answer but fails to show the derivation steps (substituting h, k, l, and r into the standard formula). Instead, it verifies a single point, which is logically insufficient to derive the equation and teaches a flawed method for finding equations of spheres.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-20 — The solution provides the correct final answer but fails to show the derivation steps (substituting h, k, l, and r into the standard formula). Instead, it verifies a single point, which is logically insufficient to derive the equation and teaches a flawed method for finding equations of spheres.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius to derive the final answer. The verification step in line 2 is valid.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution claims that the point (6, -2, -5) is on the sphere, but this point is at distance 4 from the center (2, -2, -5), so it is indeed on the sphere. However, the equation line is labeled 'unchecked' and the logic is circular or redundant rather than deriving the standard form. More critically, the prompt asks for the equation in standard form, and the solution provides a check of a single point rather than deriving the equation from the definition. The sentence in step 1 is correct, but step 2 is not a derivation of the equation, it's a verification of a specific point. A proper solution should substitute h, k, l, and r into the formula from step 1. Step 2 does not produce the final answer; it just checks a point. The stated answer is correct, but the solution steps do not logically lead to it as a derivation. It skips the actual substitution step.
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.