Home›Calculus 3›Equations of spheres›Problem 9.842

Equations of spheres

Problem 9.842 · easy

Find the equation of the sphere with center \( \displaystyle (-5, -2, 0) \) 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.
\[ \left. z^{2} + \left(x + 5\right)^{2} + \left(y + 2\right)^{2} \right|_{\substack{ x=-3\\ y=-2\\ z=0 }} = 4 \]
A point at distance r along the x-direction is on it.✓ Proved

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

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 (style) — [domain objection, downgraded to style] The solution checks a point (-3, -2, 0) which is at distance 2 from the center, but this single check does not constitute a derivation of the equation. More importantly, the problem asks for the equation, and the solution provides a verification step labeled as 'proved' without showing the actual substitution of the center coordinates into the standard formula to derive the final equation. The sentence in step 1 is correct, but step 2 is a non-sequitur as a derivation method; it verifies a specific point rather than deriving the general equation. However, looking closely at the prompt's constraints: 'Each line... is either an EQUATION... or a SENTENCE'. Step 2 is marked as an equation and 'proved'. The issue is that the solution does not actually *derive* the equation in the text provided, it just states the formula and then verifies a point. A student would not learn how to get from the center/radius to the equation. But is it 'wrong'? The final answer is correct. The 'solution' provided is incomplete/misleading as a derivation. Let's look for a harder error. Actually, the prompt asks to review the *solution*. The solution consists of a sentence defining the formula and an equation checking a point. It fails to actually perform the substitution (x - (-5))^2 etc. to get the final result. It jumps from the general formula to a specific point check. This is a logical gap. A student might think that checking one point is sufficient to define the sphere, or they might be confused about how the equation was obtained. Let's re-read carefully. Step 1: General formula. Step 2: Checks that (-3, -2, 0) is on the sphere defined by the *answer*. This is not a derivation. It's a verification of a specific point. It does not show how the answer was derived. Is there an error in the math? Center (-5, -2, 0). Radius 2. Equation: (x+5)^2 + (y+2)^2 + z^2 = 4. Point (-3, -2, 0): (-3+5)^2 + 0 + 0 = 2^2 = 4. Correct. The verdict should be 'fail' because the solution does not actually solve the problem; it only verifies a single point on the proposed solution. It teaches nothing about how to construct the equation from the center and radius. It is misleading/incomplete.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-20 — [domain objection, downgraded to style] The solution checks a point (-3, -2, 0) which is at distance 2 from the center, but this single check does not constitute a derivation of the equation. More importantly, the problem asks for the equation, and the solution provides a verification step labeled as 'proved' without showing the actual substitution of the center coordinates into the standard formula to derive the final equation. The sentence in step 1 is correct, but step 2 is a non-sequitur as a derivation method; it verifies a specific point rather than deriving the general equation. However, looking closely at the prompt's constraints: 'Each line... is either an EQUATION... or a SENTENCE'. Step 2 is marked as an equation and 'proved'. The issue is that the solution does not actually *derive* the equation in the text provided, it just states the formula and then verifies a point. A student would not learn how to get from the center/radius to the equation. But is it 'wrong'? The final answer is correct. The 'solution' provided is incomplete/misleading as a derivation. Let's look for a harder error. Actually, the prompt asks to review the *solution*. The solution consists of a sentence defining the formula and an equation checking a point. It fails to actually perform the substitution (x - (-5))^2 etc. to get the final result. It jumps from the general formula to a specific point check. This is a logical gap. A student might think that checking one point is sufficient to define the sphere, or they might be confused about how the equation was obtained. Let's re-read carefully. Step 1: General formula. Step 2: Checks that (-3, -2, 0) is on the sphere defined by the *answer*. This is not a derivation. It's a verification of a specific point. It does not show how the answer was derived. Is there an error in the math? Center (-5, -2, 0). Radius 2. Equation: (x+5)^2 + (y+2)^2 + z^2 = 4. Point (-3, -2, 0): (-3+5)^2 + 0 + 0 = 2^2 = 4. Correct. The verdict should be 'fail' because the solution does not actually solve the problem; it only verifies a single point on the proposed solution. It teaches nothing about how to construct the equation from the center and radius. It is misleading/incomplete.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius accurately. The verification step in line 2 is valid.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution checks a point (-3, -2, 0) which is at distance 2 from the center, but this is not a valid verification step because the problem asks for the equation, not a proof of a specific point. More importantly, the 'equation' line is syntactically malformed and does not represent a standard verification or derivation step. The final answer is correct, but the provided solution steps are nonsensical and do not constitute a valid derivation.
gpt-oss:20b: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.