Home›Calculus 3›Quadric surfaces in standard form›Problem 9.651

Quadric surfaces in standard form

Problem 9.651 · easy

Write \( \displaystyle - 16 x^{2} - 4 y^{2} + 16 z^{2} = 144 \) in standard form and identify the surface.

Divide through by 144.
Reviewed
\[ \frac{- 16 x^{2} - 4 y^{2} + 16 z^{2}}{144} = - \frac{x^{2}}{9} - \frac{y^{2}}{36} + \frac{z^{2}}{9} \]
The standard form.✓ Proved
A hyperboloid of two sheets: one square positive and two negative, = 1.
Reviewed

Answer \( - \frac{x^{2}}{9} - \frac{y^{2}}{36} + \frac{z^{2}}{9} = 1\ (\text{hyperboloid of two sheets}) \)

✓ 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 divides by 144 to obtain the standard form and accurately identifies the surface as a hyperboloid of two sheets based on the signs of the coefficients.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly divides by 144 to obtain the standard form and accurately identifies the surface as a hyperboloid of two sheets based on the signs of the coefficients.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — 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 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The algebraic manipulation to standard form is correct, and the identification of the surface as a hyperboloid of two sheets based on the signs of the coefficients is accurate.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.