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Equations of spheres

Problem 9.624 · easy

Find the equation of the sphere with center \( \displaystyle (-2, -1, 2) \) and radius \( \displaystyle 2 \), in standard form.

A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
\[ \left. \left(x + 2\right)^{2} + \left(y + 1\right)^{2} + \left(z - 2\right)^{2} \right|_{\substack{ x=0\\ y=-1\\ z=2 }} = 4 \]
A point at distance r along the x-direction is on it.✓ Proved

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

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 claims that the point (0, -1, 2) is on the sphere, but this point is at distance 2 from the center (-2, -1, 2), not distance 2 along the x-direction from the center in a way that validates the radius. More importantly, the equation line is syntactically nonsensical as a verification step (Subs(...)=4) and does not constitute a valid derivation or check of the final answer. The setup sentence is correct, but the 'equation' line is not a valid mathematical step to derive or verify the sphere equation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution claims that the point (0, -1, 2) is on the sphere, but this point is at distance 2 from the center (-2, -1, 2), not distance 2 along the x-direction from the center in a way that validates the radius. More importantly, the equation line is syntactically nonsensical as a verification step (Subs(...)=4) and does not constitute a valid derivation or check of the final answer. The setup sentence is correct, but the 'equation' line is not a valid mathematical step to derive or verify the sphere equation.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution claims to verify a point on the sphere, but the point (0, -1, 2) is at distance 2 from the center (-2, -1, 2), which is correct. However, the equation line is syntactically confusing and does not clearly demonstrate the standard form derivation. More importantly, the 'sentence' step defines the formula correctly, but the 'equation' step is presented as a verification of a single point rather than a derivation of the general equation. While the final answer is correct, the solution logic is flawed because it skips the actual substitution of h, k, l, and r into the formula to derive the equation, instead just checking one point. This is misleading as it implies checking one point is sufficient to find the equation.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly substitute the given center (-2, -1, 2) and radius 2 into the standard formula. Instead, it performs an unrelated check with a point (0, -1, 2) that is not derived from the problem statement, leaving the derivation of the final equation unjustified.
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.