Home›Calculus 3›Quadric surfaces in standard form›Problem 9.655

Quadric surfaces in standard form

Problem 9.655 · easy

Write \( \displaystyle 72 x^{2} + 18 y^{2} - 8 z^{2} = 0 \) in standard form and identify the surface.

Divide through by 288 and solve for the variable to the first power.
\[ \frac{72 x^{2} + 18 y^{2} - 8 z^{2}}{288} = \frac{x^{2}}{4} + \frac{y^{2}}{16} - \frac{z^{2}}{36} \]
The standard form.✓ Proved
A elliptic cone: two squares minus one, = 0.

Answer \( \frac{x^{2}}{4} + \frac{y^{2}}{16} - \frac{z^{2}}{36} = 0\ (\text{elliptic cone}) \)

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 (error) — The sentence in step 1 is mathematically nonsensical; one does not 'solve for a variable' to obtain an equation equal to zero, one simply divides by the constant term. Additionally, step 3 describes the signature as 'two squares minus one', which is vague and potentially misleading compared to the standard description of an elliptic cone (sum of two squares equals a third square).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The sentence in step 1 is mathematically nonsensical; one does not 'solve for a variable' to obtain an equation equal to zero, one simply divides by the constant term. Additionally, step 3 describes the signature as 'two squares minus one', which is vague and potentially misleading compared to the standard description of an elliptic cone (sum of two squares equals a third square).
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The setup instruction in step 1 is incorrect because the equation is homogeneous (all terms are degree 2), so there is no variable to the first power to solve for. The standard form is derived by dividing by the constant term that makes the RHS 1, but here the RHS is 0, so one simply divides by a common factor to normalize coefficients, not to isolate a linear term.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — Step 1 claims to 'solve for the variable to the first power', but no variable appears to the first power in the equation, and the subsequent step does not isolate any variable. This is a nonsensical instruction.
gpt-oss:20b: fail (error) 2026-10-14 — The division by 288 is incorrect; dividing the original equation by 72 yields x^2 + (1/4)y^2 – (1/9)z^2 = 0, not the stated standard form. The resulting equation x^2/4 + y^2/16 – z^2/36 = 0 does not represent the given surface.

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 2026-10-17 with SymPy 1.14.0.