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Equations of spheres

Problem 9.700 · easy

Find the equation of the sphere with center $ \displaystyle (-3, -1, -5) $ 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.
\[ \left. \left(x + 3\right)^{2} + \left(y + 1\right)^{2} + \left(z + 5\right)^{2} \right|_{\substack{ x=-1\\ y=-1\\ z=-5 }} = 4 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer $ \left(x + 3\right)^{2} + \left(y + 1\right)^{2} + \left(z + 5\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 substitutes the point (-1, -1, -5) into the equation, but this point is not on the sphere (distance from center (-3, -1, -5) is 2, so it is on the sphere, wait. Distance is sqrt((-1 - -3)^2 + 0 + 0) = sqrt(4) = 2. So the point IS on the sphere. However, the label says 'A point at distance r along the x-direction is on it.' The point (-1, -1, -5) is indeed at distance 2 along the x-direction from the center. But the substitution line is labeled as an 'equation, proved' by a CAS, which is confusing. More importantly, the sentence in step 1 defines the formula, but step 2 is a weird verification step rather than a derivation. The prompt asks to review the solution. Step 2 is not a standard derivation step. It's a check. But is it wrong? The point (-1, -1, -5) is on the sphere. The substitution yields 4. This is correct. But is it a good solution? It's very strange. Let's look closer. The problem asks for the equation. The answer is given. The solution has a sentence defining the formula, and then an equation that checks a point. This is not a derivation. It's a verification. Is it 'wrong'? It doesn't derive the answer. It just checks it. But the answer is already stated. The solution is incomplete as a derivation. However, the instructions say 'check that... each conclusion follows from the lines before it'. The conclusion (the answer) is not derived from the lines. The lines just define the formula and check a point. This is a 'misleading' or 'error' in logic because it doesn't actually solve the problem, it just verifies a specific point. A student would learn that checking one point is sufficient to find the equation, which is false. So 'misleading' or 'error'. Let's call it 'error' because the logic is flawed: verifying one point does not prove the equation is correct (though in this context, combined with the definition, it's weak). Actually, the most severe issue is that it doesn't show how to get the equation from the center and radius. It just states the formula and checks a point. This teaches a student that you don't need to substitute h, k, l, r into the formula, just check a point. That is false. So 'misleading'.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution substitutes the point (-1, -1, -5) into the equation, but this point is not on the sphere (distance from center (-3, -1, -5) is 2, so it is on the sphere, wait. Distance is sqrt((-1 - -3)^2 + 0 + 0) = sqrt(4) = 2. So the point IS on the sphere. However, the label says 'A point at distance r along the x-direction is on it.' The point (-1, -1, -5) is indeed at distance 2 along the x-direction from the center. But the substitution line is labeled as an 'equation, proved' by a CAS, which is confusing. More importantly, the sentence in step 1 defines the formula, but step 2 is a weird verification step rather than a derivation. The prompt asks to review the solution. Step 2 is not a standard derivation step. It's a check. But is it wrong? The point (-1, -1, -5) is on the sphere. The substitution yields 4. This is correct. But is it a good solution? It's very strange. Let's look closer. The problem asks for the equation. The answer is given. The solution has a sentence defining the formula, and then an equation that checks a point. This is not a derivation. It's a verification. Is it 'wrong'? It doesn't derive the answer. It just checks it. But the answer is already stated. The solution is incomplete as a derivation. However, the instructions say 'check that... each conclusion follows from the lines before it'. The conclusion (the answer) is not derived from the lines. The lines just define the formula and check a point. This is a 'misleading' or 'error' in logic because it doesn't actually solve the problem, it just verifies a specific point. A student would learn that checking one point is sufficient to find the equation, which is false. So 'misleading' or 'error'. Let's call it 'error' because the logic is flawed: verifying one point does not prove the equation is correct (though in this context, combined with the definition, it's weak). Actually, the most severe issue is that it doesn't show how to get the equation from the center and radius. It just states the formula and checks a point. This teaches a student that you don't need to substitute h, k, l, r into the formula, just check a point. That is false. So 'misleading'.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution substitutes the point (-1, -1, -5) into the equation, but this point is not on the sphere (its distance from the center (-3, -1, -5) is 2, so it is on the sphere, wait. Distance is sqrt((-1 - -3)^2 + 0 + 0) = sqrt(4) = 2. So the point IS on the sphere. However, the logic is flawed: checking a single point does not prove the equation is correct for the whole sphere. The standard method is to substitute the center and radius into the formula, not to verify a single point. While the final answer is correct, the reasoning presented in step 2 is logically insufficient to derive the equation, it only verifies one point. A student might think checking one point is sufficient to find the equation. Actually, looking closer, step 2 is labeled 'equation, proved' by a CAS, but the sentence describing it is missing or implied. The prompt asks to review sentences. Sentence 1 is correct. There is no explicit sentence for step 2, just an equation line. But usually, these problems require deriving the equation. If the solution relies on checking a point to 'find' the equation, that's a methodological error. However, often these solutions are just showing the result. Let's look at the structure. Step 1 gives the formula. Step 2 is an equation check. It doesn't show the substitution of h, k, l, r. It shows a point check. This is misleading because it suggests verification is derivation. But is it an error? The problem asks to 'Find' the equation. The solution provides the formula and then a check. It skips the actual derivation step of plugging in h=-3, k=-1, l=-5, r=2. This is a missing step, which makes the solution incomplete/misleading about how to find it. But strictly, is it wrong? The final answer is right. The 'sentence' part is only line 1. Line 2 is an equation. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. I only need to check sentences. Sentence 1 is correct. Is there a hidden sentence? No. So if I only check sentences, it passes? No, the instruction says 'Check that... each conclusion follows from the lines before it'. The conclusion is the final answer. The final answer is not explicitly stated as a sentence, but the 'Stated answer' is given. The solution block ends with an equation. The 'Stated answer' matches the equation in step 2 (mostly, step 2 has a weird substitution syntax). Step 2 says `Subs(..., (-1, -1, -5)) = 4`. This evaluates to `(-1+3)^2 + ... = 4` -> `4 = 4`. This is true. But it doesn't derive the general equation. It just checks a point. A student reading this would not learn how to construct the equation from center and radius. They would see a formula and a point check. This is misleading pedagogy. It fails to model the problem of 'finding' the equation. It models 'verifying' a point. Therefore, it is misleading.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution checks a point (-1, -1, -5) which is at distance 2 from the center, confirming it lies on the sphere, but this is a verification step, not a derivation. More critically, the prompt asks for the equation, and the 'solution' provided is just a check of a single point, which does not constitute finding the equation. However, looking at the provided 'Stated answer', it is correct. The issue is that the solution text provided (lines 1 and 2) does not actually derive the answer; it just states the formula and checks a point. But usually, these reviews focus on logical errors in the derivation. Let's look closer. Line 2 is an equation check. It substitutes a point into the LHS. The point (-1, -1, -5) is indeed distance 2 from (-3, -1, -5). So the check is valid. But does it 'find' the equation? No. It assumes the form and verifies a point. This is not a valid method to 'find' the equation uniquely. However, often in these datasets, 'solution' might just be the final steps. Let's look for a harder error. Actually, the standard form is $(x-h)^2 + ...$. Center $(-3, -1, -5)$. So $(x - (-3))^2 = (x+3)^2$. The stated answer is correct. The solution steps are: 1. State formula. 2. Check a point. This is not a derivation. But is it 'wrong'? It's incomplete. Let's re-read the prompt constraints. 'Find what is wrong'. If the solution doesn't actually solve the problem, it's an error. But usually, these are about mathematical mistakes. Wait, look at line 2: `Subs(..., (-1, -1, -5)) = 4`. This confirms that the point $(-1, -1, -5)$ is on the sphere defined by the LHS. It does not prove that the LHS is the *only* or *correct* equation for the sphere with center $(-3, -1, -5)$ and radius 2. It just shows one point fits. A student might think checking one point is sufficient to define the sphere, which is false. However, there is a more subtle potential error. Did the user make a sign error? Center $(-3, -1, -5)$. Formula $(x-h)^2$. $(x - (-3))^2 = (x+3)^2$. Correct. Radius 2, $r^2=4$. Correct. The 'solution' provided is extremely weak. It doesn't substitute the center and radius into the formula. It just checks a point. This is a 'misleading' or 'error' in method. It teaches that you can find an equation by checking a point, which is wrong. Let's look for a blatant error. Is the point $(-1, -1, -5)$ actually on the sphere? Distance squared: $(-1 - (-3))^2 + (-1 - (-1))^2 + (-5 - (-5))^2 = 2^2 + 0 + 0 = 4$. Yes. Is the stated answer correct? Yes. Is the solution valid? No, it doesn't derive the answer. It verifies a property. I will mark it as fail/error because the method shown (checking a single point) does not constitute finding the equation of a sphere. It is logically insufficient.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.