Home›Calculus 3›Quadric surfaces in standard form›Problem 9.830

Quadric surfaces in standard form

Problem 9.830 · easy

Write \( \displaystyle - 16 x^{2} - y^{2} + 4 z^{2} = 16 \) in standard form and identify the surface.

Divide through by 16.
Reviewed
\[ \frac{- 16 x^{2} - y^{2} + 4 z^{2}}{16} = - x^{2} - \frac{y^{2}}{16} + \frac{z^{2}}{4} \]
The standard form.✓ Proved
A hyperboloid of two sheets: one square positive and two negative, = 1.
Reviewed

Answer \( - x^{2} - \frac{y^{2}}{16} + \frac{z^{2}}{4} = 1\ (\text{hyperboloid of two sheets}) \)

Lines: 1 proved, 2 reviewed. 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	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 derives the standard form and accurately identifies the surface as a hyperboloid of two sheets based on the signs of the squared terms.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly derives the standard form and accurately identifies the surface as a hyperboloid of two sheets based on the signs of the squared terms.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The algebraic manipulation is correct, and the classification of the surface as a hyperboloid of two sheets based on the signs of the squared terms is accurate.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The classification is incorrect. The equation -x^2 - y^2/16 + z^2/4 = 1 has one positive squared term and two negative squared terms, which defines a hyperboloid of one sheet, not two sheets. A hyperboloid of two sheets requires two positive and one negative squared term (e.g., x^2 + y^2 - z^2 = 1).
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.