Home›Calculus 3›Quadric surfaces in standard form›Problem 9.648

Quadric surfaces in standard form

Problem 9.648 · easy

Write \( \displaystyle 4 x^{2} + 9 y^{2} - 36 z^{2} = 0 \) in standard form and identify the surface.

Divide through by 144 and solve for the variable to the first power.
\[ \frac{4 x^{2} + 9 y^{2} - 36 z^{2}}{144} = \frac{x^{2}}{36} + \frac{y^{2}}{16} - \frac{z^{2}}{4} \]
The standard form.✓ Proved
A elliptic cone: two squares minus one, = 0.

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

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) — 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. Step 3 incorrectly describes the signature as 'two squares minus one' (implying two positive and one negative term), but the standard form derived has two positive terms and one negative term, which is actually correct for a cone, yet the phrasing is confusing and the previous step is nonsensical.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — 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. Step 3 incorrectly describes the signature as 'two squares minus one' (implying two positive and one negative term), but the standard form derived has two positive terms and one negative term, which is actually correct for a cone, yet the phrasing is confusing and the previous step is nonsensical.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The setup in step 1 is incorrect: dividing by 144 does not yield the standard form shown in step 2 (which requires dividing by 36). Additionally, the identification in step 3 describes a hyperboloid of one sheet (two squares minus one equals a non-zero constant), not a cone (which equals 0).
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The standard form of an elliptic cone is typically written as z^2 = x^2/a^2 + y^2/b^2 or x^2/a^2 + y^2/b^2 - z^2/c^2 = 0. The provided solution divides by 144, which is incorrect; dividing by 36 yields x^2/9 + y^2/4 - z^2 = 0. The stated answer x^2/36 + y^2/16 - z^2/4 = 0 is algebraically incorrect (multiplying by 144 gives 4x^2 + 9y^2 - 36z^2, wait, 144/36=4, 144/16=9, 144/4=36. So the equation in line 2 IS correct algebraically. Let me re-read. Original: 4x^2 + 9y^2 - 36z^2 = 0. Divide by 36: x^2/9 + y^2/4 - z^2 = 0. This is a standard form. Divide by 144: x^2/36 + y^2/16 - z^2/4 = 0. This is ALSO a valid standard form representation, just with different constants a, b, c. However, the sentence says 'Divide through by 144'. Why 144? Usually, we divide by the constant term if it were equal to a constant, or we arrange it as z^2 = ... Let's look at the identification. 'A elliptic cone: two squares minus one, = 0.' This is a bit informal but correct in spirit (sum of two positive squares minus one positive square equals zero). Is there an error? If I divide 4x^2 by 144, I get x^2/36. If I divide 9y^2 by 144, I get y^2/16. If I divide -36z^2 by 144, I get -z^2/4. So the equation is correct. Is the identification correct? Yes, it is an elliptic cone. Is the sentence 'Divide through by 144 and solve for the variable to the first power' misleading? There is no variable to the first power in the equation 4x^2 + 9y^2 - 36z^2 = 0. All variables are squared. You cannot 'solve for the variable to the first power' because there isn't one. This is a significant conceptual error in the instruction/sentence. The student might be confused looking for a linear term. Therefore, the sentence is erroneous because it instructs to solve for a variable to the first power, which does not exist in this homogeneous quadratic equation.
gpt-oss:20b: pass 2026-10-14

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.