Eliminating the parameter
Problem 8.486 · medium
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)},\ y = 2 \sin{\left(t \right)} \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{x^{2}}{4} + \frac{y^{2}}{4} - 1 \right|_{\substack{ x=2 \cos{\left(t \right)}\\ y=2 \sin{\left(t \right)} }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{x^{2}}{4} + \frac{y^{2}}{4} = 1\ (\text{circle}) \)
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 incorrectly introduces a generic ellipse form with parameters h,k,a,b that are never defined. It should state cos t = x/2 and sin t = y/2, leading directly to x²/4 + y²/4 = 1. The misstatement could confuse a student about the correct substitution.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly eliminate the parameter by deriving the Cartesian equation from the parametric equations. It merely verifies that the given answer satisfies the parametric form, which does not constitute a derivation or elimination of the parameter.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to explicitly eliminate the parameter by deriving the Cartesian equation from the parametric equations. It merely verifies that the given answer satisfies the parametric form, which does not constitute a derivation or elimination of the parameter.gpt-oss:20b: fail (style) 2026-10-19 — [domain objection, downgraded to style] The first sentence incorrectly introduces a generic ellipse form with parameters h,k,a,b that are never defined. It should state cos t = x/2 and sin t = y/2, leading directly to x²/4 + y²/4 = 1. The misstatement could confuse a student about the correct substitution.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly eliminate the parameter to derive the Cartesian equation; it only verifies that the parametric equations satisfy the final result. Additionally, the setup in step 1 is presented as a general template without explicitly identifying h=0, k=0, a=2, b=2, making the logical jump to the specific equation in step 2 unjustified.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution cites a general formula for an ellipse centered at (h, k) but fails to explicitly state that h=0, k=0, a=2, and b=2 for this specific problem. This omission makes the transition from the general identity to the specific equation opaque and teaches poor modeling habits.gpt-oss:20b: pass 2026-10-18
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.