Home›Calculus 3›Quadric surfaces in standard form›Problem 9.792

Quadric surfaces in standard form

Problem 9.792 · easy

Write \( \displaystyle 450 x^{2} + 50 y^{2} - 18 z^{2} = 0 \) in standard form and identify the surface.

Divide through by 450 and solve for the variable to the first power.
\[ \frac{450 x^{2} + 50 y^{2} - 18 z^{2}}{450} = x^{2} + \frac{y^{2}}{9} - \frac{z^{2}}{25} \]
The standard form.✓ Proved
A elliptic cone: two squares minus one, = 0.

Answer \( x^{2} + \frac{y^{2}}{9} - \frac{z^{2}}{25} = 0\ (\text{elliptic cone}) \)

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: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly identifies the surface as an elliptic cone. The equation x^2 + y^2/9 - z^2/25 = 0 has two positive terms and one negative term, which defines an elliptic cone. However, the standard form of an elliptic cone is typically written as z^2/c^2 = x^2/a^2 + y^2/b^2 or x^2/a^2 + y^2/b^2 - z^2/c^2 = 0. Wait, let me re-evaluate. An elliptic cone is defined by an equation with two positive coefficients and one negative coefficient (or vice versa) equal to zero. The given equation is x^2 + y^2/9 - z^2/25 = 0. This matches the form x^2/a^2 + y^2/b^2 - z^2/c^2 = 0. So it IS an elliptic cone. Why did I think it was wrong? Let me check the coefficients again. 450x^2 + 50y^2 - 18z^2 = 0. Divide by 450: x^2 + (50/450)y^2 - (18/450)z^2 = 0 => x^2 + (1/9)y^2 - (1/25)z^2 = 0. This is indeed an elliptic cone. The sentence says 'two squares minus one, = 0'. This is a correct heuristic for a cone (one sign different from the other two). Is there a subtle error? 'A elliptic cone' should be 'An elliptic cone'. That's a style issue. But is the identification correct? Yes. Is the standard form correct? Yes. Let me look closer at the prompt's request. 'identify the surface'. The answer is 'elliptic cone'. This is correct. Why would this fail? Maybe the 'standard form' usually isolates the z term? No, equal to 0 is standard for cones. Let me re-read the sentence: 'Divide through by 450 and solve for the variable to the first power.' There are no variables to the first power in the original equation. All are squared. This instruction is nonsensical for this problem. You don't 'solve for the variable to the first power' because there isn't one. You just divide to normalize the coefficients. This is a misleading/wrong step description.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The solution incorrectly identifies the surface as an elliptic cone. The equation x^2 + y^2/9 - z^2/25 = 0 has two positive terms and one negative term, which defines an elliptic cone. However, the standard form of an elliptic cone is typically written as z^2/c^2 = x^2/a^2 + y^2/b^2 or x^2/a^2 + y^2/b^2 - z^2/c^2 = 0. Wait, let me re-evaluate. An elliptic cone is defined by an equation with two positive coefficients and one negative coefficient (or vice versa) equal to zero. The given equation is x^2 + y^2/9 - z^2/25 = 0. This matches the form x^2/a^2 + y^2/b^2 - z^2/c^2 = 0. So it IS an elliptic cone. Why did I think it was wrong? Let me check the coefficients again. 450x^2 + 50y^2 - 18z^2 = 0. Divide by 450: x^2 + (50/450)y^2 - (18/450)z^2 = 0 => x^2 + (1/9)y^2 - (1/25)z^2 = 0. This is indeed an elliptic cone. The sentence says 'two squares minus one, = 0'. This is a correct heuristic for a cone (one sign different from the other two). Is there a subtle error? 'A elliptic cone' should be 'An elliptic cone'. That's a style issue. But is the identification correct? Yes. Is the standard form correct? Yes. Let me look closer at the prompt's request. 'identify the surface'. The answer is 'elliptic cone'. This is correct. Why would this fail? Maybe the 'standard form' usually isolates the z term? No, equal to 0 is standard for cones. Let me re-read the sentence: 'Divide through by 450 and solve for the variable to the first power.' There are no variables to the first power in the original equation. All are squared. This instruction is nonsensical for this problem. You don't 'solve for the variable to the first power' because there isn't one. You just divide to normalize the coefficients. This is a misleading/wrong step description.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (misleading) 2026-10-16 — The solution incorrectly states the standard form of an elliptic cone as 'two squares minus one, = 0'. The correct form is the sum of two squared terms equal to a third squared term (e.g., x^2/a^2 + y^2/b^2 = z^2/c^2), or equivalently x^2/a^2 + y^2/b^2 - z^2/c^2 = 0. The phrasing 'minus one' is ambiguous and potentially confusing, as it might imply a constant term of -1 rather than a single negative squared variable term.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The sentence in step 1 claims to 'solve for the variable to the first power,' but the equation contains no linear terms (all variables are squared). This describes an incorrect procedure for this problem.
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.