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Equations of spheres

Problem 9.914 · easy

Find the equation of the sphere with center \( \displaystyle (1, -3, 5) \) 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.
\[ \left. \left(x - 1\right)^{2} + \left(y + 3\right)^{2} + \left(z - 5\right)^{2} \right|_{\substack{ x=5\\ y=-3\\ z=5 }} = 16 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer \( \left(x - 1\right)^{2} + \left(y + 3\right)^{2} + \left(z - 5\right)^{2} = 16 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution checks a point (5, -3, 5) which is distance 4 from the center, but it does not verify that the derived equation is the general equation for the sphere. More critically, the 'equation' line is malformed and does not constitute a valid proof step; it substitutes a specific point into an expression without equating it to the radius squared in a way that validates the general form. The sentence setup is correct, but the verification step is logically insufficient and syntactically broken.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution checks a point (5, -3, 5) which is distance 4 from the center, but it does not verify that the derived equation is the general equation for the sphere. More critically, the 'equation' line is malformed and does not constitute a valid proof step; it substitutes a specific point into an expression without equating it to the radius squared in a way that validates the general form. The sentence setup is correct, but the verification step is logically insufficient and syntactically broken.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: fail (style) 2026-11-07 — [domain objection, downgraded to style] The solution incorrectly substitutes the point (5, -3, 5) into the equation. The center is (1, -3, 5), so a point at distance 4 along the x-direction would be (5, -3, 5) only if the radius was 4 and the center's x-coordinate was 1 (1+4=5). However, the substitution line claims to verify a point on the sphere but uses the center's y and z coordinates correctly while the x-coordinate calculation is coincidentally correct for the radius, but the logic is flawed because it doesn't derive the equation, it just checks a point. More importantly, the problem asks for the equation, and the solution provides the correct final answer but the 'proof' step is nonsensical as a derivation. The sentence in step 1 is correct, but step 2 is not a valid derivation step for finding the equation; it's a verification of a specific point. The prompt asks to review the solution. The solution jumps from the formula to a specific point check without showing the substitution of h, k, l, r. The equation line is marked 'proved' by a CAS, but the sentence describing it is misleading/wrong in context of deriving the answer. Actually, looking closely: Step 1 states the formula. Step 2 is an equation that checks if (5, -3, 5) is on the sphere defined by center (1, -3, 5) and radius 4. (5-1)^2 + (-3+3)^2 + (5-5)^2 = 16 + 0 + 0 = 16. This is true. But this does not constitute finding the equation. It's a non-sequitur. The student needs to substitute h=1, k=-3, l=5, r=4 into the formula. The solution fails to show this step, instead showing a point check. This is a logical gap/error in the solution process.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: fail (style) 2026-11-07 — [domain objection, downgraded to style] The solution fails to substitute the radius squared (16) into the standard equation, leaving the right-hand side as an undefined variable or implicit value in the derivation steps. Additionally, the unchecked equation line is syntactically and logically incoherent, attempting to substitute coordinates into an expression rather than verifying a point on the sphere.
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.