Home›Calculus 3›Quadric surfaces in standard form›Problem 9.737

Quadric surfaces in standard form

Problem 9.737 · easy

Write \( \displaystyle - 3 x^{2} + 3 y^{2} - 12 z = 0 \) in standard form and identify the surface.

Divide through by 12 and solve for the variable to the first power.
Reviewed
\[ \frac{- 3 x^{2} + 3 y^{2} - 12 z}{12} = - \frac{x^{2}}{4} + \frac{y^{2}}{4} - z \]
The standard form.✓ Proved
A hyperbolic paraboloid: z to the first power equals a difference of squares.
Reviewed

Answer \( - \frac{x^{2}}{4} + \frac{y^{2}}{4} - z = 0\ (\text{hyperbolic paraboloid}) \)

✓ 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 standard form and the surface type. The steps are logically sound and the conclusion follows directly from the algebraic manipulation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the standard form and the surface type. The steps are logically sound and the conclusion follows directly from the algebraic manipulation.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the surface as a hyperbolic paraboloid and derives the standard form accurately.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly divides by 12 to isolate the linear term z, resulting in the standard form z = y^2/4 - x^2/4. The identification of the surface as a hyperbolic paraboloid is correct.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.