Eliminating the parameter
Problem 8.220 · medium
Eliminate the parameter from \( \displaystyle x = 3 \cos{\left(t \right)},\ y = 3 \sin{\left(t \right)} + 1 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{x^{2}}{9} + \frac{\left(y - 1\right)^{2}}{9} - 1 \right|_{\substack{ x=3 \cos{\left(t \right)}\\ y=3 \sin{\left(t \right)} + 1 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{x^{2}}{9} + \frac{\left(y - 1\right)^{2}}{9} = 1\ (\text{circle}) \)
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 generic parameters a, b, h, k without relating them to the given equations. It should state the specific substitutions cos t = x/3 and sin t = (y−1)/3, not a general form.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly eliminate the parameter by substituting the given expressions for x and y into the identity. It states a general formula but does not show the derivation of the specific equation from the given parametric equations, leaving the connection between the problem statement and the final answer logically incomplete.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to explicitly eliminate the parameter by substituting the given expressions for x and y into the identity. It states a general formula but does not show the derivation of the specific equation from the given parametric equations, leaving the connection between the problem statement and the final answer logically incomplete.gpt-oss:20b: fail (error) 2026-10-17 — The first sentence introduces generic parameters a, b, h, k without relating them to the given equations. It should state the specific substitutions cos t = x/3 and sin t = (y−1)/3, not a general form.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly perform the elimination step (substituting the expressions for cos(t) and sin(t) derived from x and y into the identity). It jumps from stating the identity to verifying the final equation, omitting the logical deduction required to 'eliminate the parameter'.gpt-oss:20b: fail (style) 2026-10-15 — [domain objection, downgraded to style] The first sentence introduces generic parameters h,k,a,b that are not defined or related to the given equations. It does not correctly express the relationships cos t = x/3 and sin t = (y−1)/3, so the setup is incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly identify the parameters h, k, a, and b from the given equations before applying the identity. It states a generic formula but does not show the necessary substitution step (identifying a=3, b=3, h=0, k=1) that justifies the final equation, making the logical jump from the parametric form to the Cartesian form incomplete.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.