Home›Calculus 3›Quadric surfaces in standard form›Problem 9.834

Quadric surfaces in standard form

Problem 9.834 · easy

Write \( \displaystyle - 3 x^{2} + 108 y^{2} - 108 z = 0 \) in standard form and identify the surface.

Divide through by 108 and solve for the variable to the first power.
Reviewed
\[ \frac{- 3 x^{2} + 108 y^{2} - 108 z}{108} = - \frac{x^{2}}{36} + y^{2} - z \]
The standard form.✓ Proved
A hyperbolic paraboloid: z to the first power equals a difference of squares.
Reviewed

Answer \( - \frac{x^{2}}{36} + y^{2} - z = 0\ (\text{hyperbolic paraboloid}) \)

✓ 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 identifies the surface as a hyperbolic paraboloid and derives the standard form accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly identifies the surface as a hyperbolic paraboloid and derives the standard form accurately.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly manipulates the equation into standard form and accurately identifies the surface as a hyperbolic paraboloid based on the resulting structure.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the surface as a hyperbolic paraboloid and derives the standard form. The reasoning is sound.
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.