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Equations of spheres

Problem 9.616 · easy

Find the equation of the sphere with center \( \displaystyle (2, 5, 4) \) 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.
\[ \left. \left(x - 2\right)^{2} + \left(y - 5\right)^{2} + \left(z - 4\right)^{2} \right|_{\substack{ x=7\\ y=5\\ z=4 }} = 25 \]
A point at distance r along the x-direction is on it.✓ Proved

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

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 (style) — [domain objection, downgraded to style] The solution incorrectly identifies the point (7, 5, 4) as being at distance r from the center (2, 5, 4). The distance is actually 5, which matches the radius, but the equation provided in step 2 substitutes (7,5,4) into the left-hand side and claims it equals 25. While (7-2)^2 = 25 is true, the phrasing 'A point at distance r along the x-direction is on it' is a verification step, not a derivation. More critically, the problem asks for the equation, and the solution provides the correct final answer but the intermediate 'equation' line is a specific check, not the general derivation. However, the main issue is that the solution is incomplete as a derivation; it states the formula and then checks a point. It does not actually derive the equation from the center and radius given, it just asserts the formula and verifies one point. But wait, the prompt says 'Find the equation'. The solution gives the formula and then a check. It doesn't explicitly substitute h=2, k=5, l=4, r=5 into the formula to get the final answer in the text, although the 'Stated answer' is correct. The 'equation' line is just a check. The sentence in step 1 is correct. The 'equation' in step 2 is a valid check. Is it a valid solution? It's very sparse. But is it wrong? No. Is it misleading? It implies that checking one point is sufficient to define the sphere, which is false. A sphere is defined by center and radius. Checking one point doesn't prove the equation is correct for all points. It's a weak verification. However, the user asks to review the solution. The solution consists of a sentence defining the sphere and an equation checking a point. It does not explicitly show the substitution of the given center and radius into the standard form to arrive at the final answer. It jumps from the general formula to a specific check. This is a 'style' or 'misleading' issue because it doesn't show the actual work of plugging in the numbers. But looking closer at step 2: `Subs((x - 2)**2 + (y - 5)**2 + (z - 4)**2, (x, y, z), (7, 5, 4)) = 25`. This equation is true. The sentence says 'A point at distance r along the x-direction is on it.' This is true. But does this constitute a solution? No. It's a verification. The problem asks to 'Find' the equation. The solution doesn't find it, it assumes the form and checks a point. This is a logical gap. It's 'misleading' because it suggests verification is derivation. Or 'error' because it fails to perform the required task. Let's call it 'error' because the solution does not actually derive the equation from the given parameters; it merely states the general form and verifies a single point, which is not a valid method for finding the equation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution incorrectly identifies the point (7, 5, 4) as being at distance r from the center (2, 5, 4). The distance is actually 5, which matches the radius, but the equation provided in step 2 substitutes (7,5,4) into the left-hand side and claims it equals 25. While (7-2)^2 = 25 is true, the phrasing 'A point at distance r along the x-direction is on it' is a verification step, not a derivation. More critically, the problem asks for the equation, and the solution provides the correct final answer but the intermediate 'equation' line is a specific check, not the general derivation. However, the main issue is that the solution is incomplete as a derivation; it states the formula and then checks a point. It does not actually derive the equation from the center and radius given, it just asserts the formula and verifies one point. But wait, the prompt says 'Find the equation'. The solution gives the formula and then a check. It doesn't explicitly substitute h=2, k=5, l=4, r=5 into the formula to get the final answer in the text, although the 'Stated answer' is correct. The 'equation' line is just a check. The sentence in step 1 is correct. The 'equation' in step 2 is a valid check. Is it a valid solution? It's very sparse. But is it wrong? No. Is it misleading? It implies that checking one point is sufficient to define the sphere, which is false. A sphere is defined by center and radius. Checking one point doesn't prove the equation is correct for all points. It's a weak verification. However, the user asks to review the solution. The solution consists of a sentence defining the sphere and an equation checking a point. It does not explicitly show the substitution of the given center and radius into the standard form to arrive at the final answer. It jumps from the general formula to a specific check. This is a 'style' or 'misleading' issue because it doesn't show the actual work of plugging in the numbers. But looking closer at step 2: `Subs((x - 2)**2 + (y - 5)**2 + (z - 4)**2, (x, y, z), (7, 5, 4)) = 25`. This equation is true. The sentence says 'A point at distance r along the x-direction is on it.' This is true. But does this constitute a solution? No. It's a verification. The problem asks to 'Find' the equation. The solution doesn't find it, it assumes the form and checks a point. This is a logical gap. It's 'misleading' because it suggests verification is derivation. Or 'error' because it fails to perform the required task. Let's call it 'error' because the solution does not actually derive the equation from the given parameters; it merely states the general form and verifies a single point, which is not a valid method for finding the equation.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly substitute the radius r=5 into the standard equation to derive the final answer. It only verifies a single point on the sphere, which does not constitute a derivation of the equation.
gpt-oss:20b: fail (error) 2026-10-15 — The solution only verifies the equation at a single point (7,5,4) but never derives the general equation from the definition. It does not show that every point satisfying the equation is at distance 5 from the center, nor that the given point lies on the sphere. The conclusion is unsupported.
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution includes an unchecked equation that is mathematically false: substituting the point (7, 5, 4) into the left-hand side yields 25, but the equation claims this equals 25 without showing the substitution result or verifying the point lies on the sphere. More critically, the line is marked 'unchecked' yet presented as a step in the derivation, which is logically invalid. The standard form is correct, but the supporting 'verification' step is presented as a proven fact when it is neither checked nor derived properly.
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.