Home›Calculus 3›Quadric surfaces in standard form›Problem 9.790

Quadric surfaces in standard form

Problem 9.790 · easy

Write \( \displaystyle 72 x^{2} + 50 y^{2} - 50 z^{2} = 1800 \) in standard form and identify the surface.

Divide through by 1800.
Reviewed
\[ \frac{72 x^{2} + 50 y^{2} - 50 z^{2}}{1800} = \frac{x^{2}}{25} + \frac{y^{2}}{36} - \frac{z^{2}}{36} \]
The standard form.✓ Proved
A hyperboloid of one sheet: two squares positive and one negative, = 1.
Reviewed

Answer \( \frac{x^{2}}{25} + \frac{y^{2}}{36} - \frac{z^{2}}{36} = 1\ (\text{hyperboloid of one sheet}) \)

✓ 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 the constant term to obtain the standard form and accurately identifies the surface based on the signs of the squared terms.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly divides by the constant term to obtain the standard form and accurately identifies the surface based on the signs of the squared terms.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The algebraic manipulation is correct, and the classification of the quadric surface based on the signs of the coefficients is accurate.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The algebraic manipulation to standard form is correct, and the classification of the surface as a hyperboloid of one sheet based on the signs of the coefficients is accurate.
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 2026-10-19 with SymPy 1.14.0.