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Quadric surfaces in standard form

Problem 9.837 · easy

Write \( \displaystyle 1296 x^{2} + 2025 y^{2} + 3600 z^{2} = 32400 \) in standard form and identify the surface.

Divide through by 32400.
Reviewed
\[ \frac{1296 x^{2} + 2025 y^{2} + 3600 z^{2}}{32400} = \frac{x^{2}}{25} + \frac{y^{2}}{16} + \frac{z^{2}}{9} \]
The standard form.✓ Proved
A ellipsoid: x²/a² + y²/b² + z²/c² = 1.
Reviewed

Answer \( \frac{x^{2}}{25} + \frac{y^{2}}{16} + \frac{z^{2}}{9} = 1\ (\text{ellipsoid}) \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the surface classified from the signs of its coefficients; the standard form is a multiple of the given equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly normalizes the equation to standard form and accurately identifies the surface as an ellipsoid.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly normalizes the equation to standard form and accurately identifies the surface as an ellipsoid.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly divides by the constant term to achieve standard form and accurately identifies the surface as an ellipsoid.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-19 — The article 'A' is incorrect before 'ellipsoid'; it should be 'An ellipsoid'. While the math is correct, the grammar error is notable.
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/quadric_surfaces, checked 2026-10-20 with SymPy 1.14.0.