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Eliminating the parameter

Problem 8.227 · medium

Eliminate the parameter from \( \displaystyle x = 3 \cos{\left(t \right)},\ y = 3 \sin{\left(t \right)} - 2 \), and say what kind of curve it is.

cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
\[ \left. \frac{x^{2}}{9} + \frac{\left(y + 2\right)^{2}}{9} - 1 \right|_{\substack{ x=3 \cos{\left(t \right)}\\ y=3 \sin{\left(t \right)} - 2 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \frac{x^{2}}{9} + \frac{\left(y + 2\right)^{2}}{9} = 1\ (\text{circle}) \)

Lines: 1 proved, 1 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	points on the parametric curve at four values of t satisfy the equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The sentence in step 1 is mathematically imprecise and potentially misleading. It states 'cos t = (x − h)/a' as a general identity, but h, a, and b are not defined in the context of the specific problem until they are implicitly derived. A student might not know how to determine h, k, a, and b from the given parametric equations. The step should explicitly identify the center and radius from the given equations (x = 3 cos t, y = 3 sin t - 2) before stating the substitution.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The sentence in step 1 is mathematically imprecise and potentially misleading. It states 'cos t = (x − h)/a' as a general identity, but h, a, and b are not defined in the context of the specific problem until they are implicitly derived. A student might not know how to determine h, k, a, and b from the given parametric equations. The step should explicitly identify the center and radius from the given equations (x = 3 cos t, y = 3 sin t - 2) before stating the substitution.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly eliminate the parameter to derive the Cartesian equation, instead only verifying that the parametric equations satisfy the final result. It also incorrectly states the general form as cos t = (x - h)/a without defining h, k, a, or b in the context of the specific problem.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly identify the center (h, k) and radius a, b from the given parametric equations before applying the identity. It jumps to a generic formula without showing how x=3cos(t) and y=3sin(t)-2 map to it, leaving the derivation of the specific constants (h=0, k=-2, a=3, b=3) implicit and unverified.
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/parametric_eliminate, checked 2026-10-17 with SymPy 1.14.0.