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Eliminating the parameter

Problem 8.358 · medium

Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)} - 1,\ y = 5 \sin{\left(t \right)} + 3 \), and say what kind of curve it is.

cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
\[ \left. \frac{\left(x + 1\right)^{2}}{25} + \frac{\left(y - 3\right)^{2}}{25} - 1 \right|_{\substack{ x=5 \cos{\left(t \right)} - 1\\ y=5 \sin{\left(t \right)} + 3 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \frac{\left(x + 1\right)^{2}}{25} + \frac{\left(y - 3\right)^{2}}{25} = 1\ (\text{circle}) \)

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 solution fails to explicitly solve for cos(t) and sin(t) in terms of x and y, which is the core task of eliminating the parameter. It merely verifies that the parametric equations satisfy the final Cartesian equation, but does not demonstrate the derivation or address the domain/range constraints inherent in the elimination process.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution fails to explicitly solve for cos(t) and sin(t) in terms of x and y, which is the core task of eliminating the parameter. It merely verifies that the parametric equations satisfy the final Cartesian equation, but does not demonstrate the derivation or address the domain/range constraints inherent in the elimination process.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to explicitly identify the center and radius from the derived equation, which is necessary to fully answer 'what kind of curve it is' (specifically, a circle with center (-1, 3) and radius 5). Merely stating 'circle' is incomplete without these parameters.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The sentence in step 1 incorrectly states the substitution formulas as cos t = (x - h)/a and sin t = (y - k)/b, which implies x = h + a cos t. However, the given equation is x = 5 cos(t) - 1, which corresponds to h = -1, not h = 1. The correct substitution should be cos t = (x + 1)/5. While the final equation is correct, the stated method contains a sign error that would mislead a student attempting to follow the logic.
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/parametric_eliminate, checked 2026-10-18 with SymPy 1.14.0.