Eliminating the parameter
Problem 8.346 · medium
Eliminate the parameter from \( \displaystyle x = 3 \cos{\left(t \right)} + 1,\ y = 3 \sin{\left(t \right)} + 2 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{\left(x - 1\right)^{2}}{9} + \frac{\left(y - 2\right)^{2}}{9} - 1 \right|_{\substack{ x=3 \cos{\left(t \right)} + 1\\ y=3 \sin{\left(t \right)} + 2 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{\left(x - 1\right)^{2}}{9} + \frac{\left(y - 2\right)^{2}}{9} = 1\ (\text{circle}) \)
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: fail (error) — The first sentence introduces variables h, k, a, b without defining them. It should state that cos t = (x-1)/3 and sin t = (y-2)/3, then use cos²t+sin²t=1. As written, the setup is incomplete and misleading.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly eliminate the parameter by deriving the Cartesian equation from the given 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 the problem.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly eliminate the parameter by deriving the Cartesian equation from the given 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 the problem.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence introduces variables h, k, a, b without defining them. It should state that cos t = (x-1)/3 and sin t = (y-2)/3, then use cos²t+sin²t=1. As written, the setup is incomplete and misleading.qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution fails to actually eliminate the parameter to derive the Cartesian equation; it only verifies that the parametric equations satisfy the final answer. Furthermore, the first sentence introduces undefined variables h, a, and b without linking them to the specific problem values.gpt-oss:20b: fail (misleading) 2026-10-16 — The first sentence introduces variables h, k, a, b without defining them, and does not explicitly relate them to the given parametrization. It therefore does not correctly model the problem and could mislead a student about how to eliminate the parameter.qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution uses a generic template with undefined variables h, a, and b without explicitly identifying them from the given equations. It fails to show the necessary algebraic steps of isolating cos(t) and sin(t) from the specific parametric equations provided.gpt-oss:20b: fail (misleading) 2026-10-14 — The first sentence introduces variables h, k, a, b without defining them, and does not explicitly relate them to the given parametric equations. It therefore fails to correctly model the problem and could mislead a student into thinking any circle of the form (x−h)²/a²+(y−k)²/b²=1 satisfies the given parametrization. The subsequent equation is correct, but the initial statement is incomplete and potentially confusing.
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.