Home›Calculus 3›Quadric surfaces in standard form›Problem 9.793

Quadric surfaces in standard form

Problem 9.793 · easy

Write \( \displaystyle - 75 x^{2} + 12 y^{2} - 300 z = 0 \) in standard form and identify the surface.

Divide through by 300 and solve for the variable to the first power.
\[ \frac{- 75 x^{2} + 12 y^{2} - 300 z}{300} = - \frac{x^{2}}{4} + \frac{y^{2}}{25} - z \]
The standard form.✓ Proved
A hyperbolic paraboloid: z to the first power equals a difference of squares.

Answer \( - \frac{x^{2}}{4} + \frac{y^{2}}{25} - z = 0\ (\text{hyperbolic paraboloid}) \)

Lines: 1 proved, 2 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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: fail (error) — Sentence 1 incorrectly states that the equation is solved for the variable to the first power; the step only divides by 300, not isolate z. The rest is correct.
qwen3.6:27b-mlx: pass — The solution correctly derives the standard form and accurately identifies the surface as a hyperbolic paraboloid based on the linear term in z and the quadratic terms in x and y with opposite signs.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly derives the standard form and accurately identifies the surface as a hyperbolic paraboloid based on the linear term in z and the quadratic terms in x and y with opposite signs.
gpt-oss:20b: fail (error) 2026-10-19 — Sentence 1 incorrectly states that the equation is solved for the variable to the first power; the step only divides by 300, not isolate z. The rest is correct.
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly derives the standard form and accurately identifies the surface as a hyperbolic paraboloid.
gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence claims to solve for the variable to the first power, but only divides by 300 without isolating z. This misleads the reader about the algebraic manipulation performed.

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 2026-10-19 with SymPy 1.14.0.