Home›Calculus 3›Quadric surfaces in standard form›Problem 9.735

Quadric surfaces in standard form

Problem 9.735 · easy

Write \( \displaystyle 8 x^{2} + 2 y^{2} - 72 z = 0 \) in standard form and identify the surface.

Divide through by 72 and solve for the variable to the first power.
Reviewed
\[ \frac{8 x^{2} + 2 y^{2} - 72 z}{72} = \frac{x^{2}}{9} + \frac{y^{2}}{36} - z \]
The standard form.✓ Proved
A elliptic paraboloid: z (or another variable) to the first power equals a sum of squares.
Reviewed

Answer \( \frac{x^{2}}{9} + \frac{y^{2}}{36} - z = 0\ (\text{elliptic paraboloid}) \)

✓ 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 manipulates the equation into standard form and accurately identifies the surface as an elliptic paraboloid based on the presence of one linear variable and two quadratic variables with the same sign.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly manipulates the equation into standard form and accurately identifies the surface as an elliptic paraboloid based on the presence of one linear variable and two quadratic variables with the same sign.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly manipulates the equation into standard form and accurately identifies the surface type based on the resulting structure.
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 based on the equation structure.
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.