Eliminating the parameter
Problem 8.489 · medium
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} + 2,\ y = 2 \sin{\left(t \right)} - 2 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{\left(x - 2\right)^{2}}{4} + \frac{\left(y + 2\right)^{2}}{4} - 1 \right|_{\substack{ x=2 \cos{\left(t \right)} + 2\\ y=2 \sin{\left(t \right)} - 2 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{\left(x - 2\right)^{2}}{4} + \frac{\left(y + 2\right)^{2}}{4} = 1\ (\text{circle}) \)
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 (style) — [domain objection, downgraded to style] The first sentence introduces a generic relation with symbols h, k, a, b that are never defined or used later. It is unclear how it applies to the given parametrization, which would be clearer if it explicitly stated h=2, k=-2, a=2, b=2.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly derive the Cartesian equation from the parametric equations, instead stating a generic identity and verifying the final answer. It does not show the necessary algebraic steps (isolating cos(t) and sin(t)) to actually 'eliminate the parameter' as requested.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to explicitly derive the Cartesian equation from the parametric equations, instead stating a generic identity and verifying the final answer. It does not show the necessary algebraic steps (isolating cos(t) and sin(t)) to actually 'eliminate the parameter' as requested.gpt-oss:20b: fail (style) 2026-10-19 — [domain objection, downgraded to style] The first sentence introduces a generic relation with symbols h, k, a, b that are never defined or used later. It is unclear how it applies to the given parametrization, which would be clearer if it explicitly stated h=2, k=-2, a=2, b=2.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly perform the algebraic elimination steps (isolating cos(t) and sin(t) from the given parametric equations) required by the problem statement. It merely states a general identity and verifies the final answer, skipping the actual derivation.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence states a generic relation cos t = (x−h)/a and sin t = (y−k)/b without specifying the actual values h = 2, k = −2, a = b = 2. This vague wording could mislead a student into thinking the parameters are arbitrary, whereas the problem requires the specific substitution for this parametrization.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly perform the algebraic elimination steps (isolating cos(t) and sin(t) and substituting them into the identity) required by the problem statement. It merely verifies that the final equation is satisfied, which is a check, not a derivation.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence is too generic and does not correctly identify the specific values of h, k, a, and b for the given parametrization. It could mislead a student into applying the wrong substitutions.
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 2026-10-19 with SymPy 1.14.0.