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Equations of spheres

Problem 9.913 · easy

Find the equation of the sphere with center \( \displaystyle (0, 3, -4) \) 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.
\[ \left. x^{2} + \left(y - 3\right)^{2} + \left(z + 4\right)^{2} \right|_{\substack{ x=4\\ y=3\\ z=-4 }} = 16 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer \( x^{2} + \left(y - 3\right)^{2} + \left(z + 4\right)^{2} = 16 \)

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: fail (error) — The solution only plugs a single point into the general sphere equation; it does not justify that the resulting equation represents all points at distance 4 from the center (0,3,−4). The sentence claiming a point at distance r along the x‑direction is on it is irrelevant and does not establish the standard form.
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly substitute the given center coordinates (0, 3, -4) into the standard equation formula. Instead, it verifies a single point, which is not a derivation of the equation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution fails to explicitly substitute the given center coordinates (0, 3, -4) into the standard equation formula. Instead, it verifies a single point, which is not a derivation of the equation.
gpt-oss:20b: fail (error) 2026-11-08 — The solution only plugs a single point into the general sphere equation; it does not justify that the resulting equation represents all points at distance 4 from the center (0,3,−4). The sentence claiming a point at distance r along the x‑direction is on it is irrelevant and does not establish the standard form.
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius accurately. The verification step, while unconventional, is mathematically valid.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: fail (error) 2026-11-07 — The solution uses a nonsensical verification step involving a CAS substitution command that does not logically follow from the setup or prove the general equation. It fails to demonstrate the derivation of the standard form from the given center and radius.
gpt-oss:20b: fail (error) 2026-11-07 — The solution substitutes the point (4,3,-4) into the equation, which is not the center of the sphere (the center is (0,3,-4)). Moreover, the sentence claiming that a point at distance r along the x‑direction is on the sphere is misleading and does not justify the equation. The correct approach is to write (x−0)²+(y−3)²+(z+4)²=4² and no substitution is needed.

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 2026-11-08 with SymPy 1.14.0.