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Eliminating the parameter

Problem 8.369 · medium

Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} + 1,\ y = 4 \sin{\left(t \right)} + 1 \), and say what kind of curve it is.

cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
\[ \left. \left(x - 1\right)^{2} + \frac{\left(y - 1\right)^{2}}{16} - 1 \right|_{\substack{ x=\cos{\left(t \right)} + 1\\ y=4 \sin{\left(t \right)} + 1 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \left(x - 1\right)^{2} + \frac{\left(y - 1\right)^{2}}{16} = 1\ (\text{ellipse}) \)

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 (error) — The solution fails to explicitly derive the Cartesian equation from the parametric equations. It states a general identity and verifies the final answer, but skips the crucial algebraic steps of isolating cos(t) and sin(t) and applying the Pythagorean identity, which is the core task of 'eliminating the parameter'.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly derive the Cartesian equation from the parametric equations. It states a general identity and verifies the final answer, but skips the crucial algebraic steps of isolating cos(t) and sin(t) and applying the Pythagorean identity, which is the core task of 'eliminating the parameter'.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the curve as an ellipse and provides the correct Cartesian equation. The verification step confirms the parametric equations satisfy the derived implicit equation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly derive the Cartesian equation from the parametric equations, instead stating a generic identity and then verifying the final answer. It skips the necessary algebraic steps (isolating cos(t) and sin(t)) required to 'eliminate the parameter' as requested by the problem statement.
gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence introduces symbols h,k,a,b without defining them, and incorrectly states the relations for cos t and sin t. It should explicitly write cos t = x−1 and sin t = (y−1)/4, then use the identity cos²t+sin²t=1 to eliminate the parameter. The current wording could mislead a student into thinking the parameters are arbitrary.

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.