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Equations of spheres

Problem 9.911 · easy

Find the equation of the sphere with center \( \displaystyle (-4, 2, 1) \) and radius \( \displaystyle 1 \), in standard form.

A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
\[ \left. \left(x + 4\right)^{2} + \left(y - 2\right)^{2} + \left(z - 1\right)^{2} \right|_{\substack{ x=-3\\ y=2\\ z=1 }} = 1 \]
A point at distance r along the x-direction is on it.✓ Proved

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

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 checks a point (-3, 2, 1) which is at distance 1 from the center, but the problem asks for the equation of the sphere, not a verification of a single point. The stated answer is correct, but the provided 'solution' steps do not constitute a derivation of the equation; they merely verify a property of the result using an irrelevant check that doesn't prove the equation is the *only* or *correct* one for the given parameters. More critically, the prompt asks to review the *solution* provided. The solution consists of a definition and a check. It does not actually derive the equation. However, usually, these tasks imply the 'Stated answer' is the result of the solution. If the solution is just these two lines, it fails to derive the answer. But looking closer at the prompt structure: 'Stated answer' is separate. The 'Solution' is the work. The work is insufficient to derive the answer. It just states the formula and checks a point. This is not a valid derivation. Furthermore, checking one point does not prove the equation is correct for all points. It's a logical gap. But is it an 'error' in the solution's logic or just incomplete? The prompt asks to find what is wrong. The solution does not actually produce the equation. It just asserts the formula and checks a point. This is not a solution. It's a partial check. Therefore, it fails to solve the problem.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution checks a point (-3, 2, 1) which is at distance 1 from the center, but the problem asks for the equation of the sphere, not a verification of a single point. The stated answer is correct, but the provided 'solution' steps do not constitute a derivation of the equation; they merely verify a property of the result using an irrelevant check that doesn't prove the equation is the *only* or *correct* one for the given parameters. More critically, the prompt asks to review the *solution* provided. The solution consists of a definition and a check. It does not actually derive the equation. However, usually, these tasks imply the 'Stated answer' is the result of the solution. If the solution is just these two lines, it fails to derive the answer. But looking closer at the prompt structure: 'Stated answer' is separate. The 'Solution' is the work. The work is insufficient to derive the answer. It just states the formula and checks a point. This is not a valid derivation. Furthermore, checking one point does not prove the equation is correct for all points. It's a logical gap. But is it an 'error' in the solution's logic or just incomplete? The prompt asks to find what is wrong. The solution does not actually produce the equation. It just asserts the formula and checks a point. This is not a solution. It's a partial check. Therefore, it fails to solve the problem.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: fail (error) 2026-11-07 — The solution claims that the point (-3, 2, 1) is on the sphere, but this point is at distance 1 from the center (-4, 2, 1) only if the radius is 1. Wait, distance is sqrt((-3 - -4)^2) = 1. So the point IS on the sphere. However, the equation line is labeled 'proved' by a CAS, but the sentence setup is just the definition. The issue is that the solution doesn't actually derive the answer, it just states the definition and then checks a single point. But more importantly, the prompt asks for the equation. The solution provides the definition and a check. It doesn't explicitly state the final equation as a derived result, although the stated answer is correct. Let's look closer. The solution has 2 lines. Line 1 is the formula. Line 2 is a check. It never actually writes down the final equation `(x+4)^2 + ... = 1` as a derived line, it just checks a point. But the 'Stated answer' is separate. The review is of the 'Solution' lines. Line 2 is an equation marked 'proved'. It checks a point. This is not a derivation. It's a verification. Is it 'wrong'? It's incomplete. But is it 'error'? The problem asks to 'Find the equation'. The solution provides a definition and a check. It fails to actually produce the equation as a logical conclusion. However, usually these reviews look for mathematical errors. Let's look at the point again. Center (-4, 2, 1). Point (-3, 2, 1). Distance is 1. Radius is 1. So the point is on the sphere. The check is valid. But the solution is logically insufficient to 'find' the equation. It just asserts the form and checks one point. A student might think checking one point is enough to define the sphere. This is misleading. Also, the variable `l` in line 1 is used for z-coordinate, which is non-standard (usually `c` or `z_0`), but acceptable. The main issue is that the solution does not actually derive the equation, it just states the general form and verifies a point. This is not a valid derivation. It's a 'style' or 'misleading' issue because it teaches that checking one point is sufficient to determine the equation, which is false (you need the center and radius, or multiple points). Actually, the center is given. So you just plug in. The solution skips the 'plug in' step and goes straight to a check. This is misleading.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: fail (error) 2026-11-07 — The solution checks the point (-3, 2, 1), which is at distance 1 from the center (-4, 2, 1). However, the problem asks for the equation of the sphere, not a verification of a single point. While the final answer is correct, the provided 'solution' steps are logically insufficient to derive the equation and the second step is a verification, not a derivation. More critically, the prompt implies reviewing the *logic* of the solution steps. Step 2 is labeled 'unchecked' but claims to verify a point. If this is meant to be the derivation, it is non-sequitur. If it is meant to be a check, it is incomplete. However, looking closely at the provided text, Step 2 is an equation that is mathematically true for the given answer, but it does not constitute a solution method. The primary issue is that the 'solution' provided is just a definition and a check, lacking the actual substitution step that derives the answer from the definition. But wait, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Step 2 is marked [equation, unchecked]. The system says 'do not re-check the algebra of a line marked proved', but this is marked 'unchecked'. Actually, the prompt says 'computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. Step 2 is NOT marked proved. It is marked 'unchecked'. This implies I should check it? Or is it a label from the user? Let's look at the content. Step 2 substitutes (-3, 2, 1) into the LHS. (-3+4)^2 + (2-2)^2 + (1-1)^2 = 1^2 + 0 + 0 = 1. This is correct. But does this solve the problem? No. The problem asks to *find* the equation. The solution provides the definition and then checks a point. It never actually performs the substitution of h=-4, k=2, l=1, r=1 into the formula. Therefore, the solution is incomplete/incorrect as a derivation. It fails to show the work required to get the answer.
gpt-oss:20b: pass 2026-11-07

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.