Eliminating the parameter
Problem 8.490 · medium
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} + 1,\ y = \sin{\left(t \right)} - 2 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \left(x - 1\right)^{2} + \left(y + 2\right)^{2} - 1 \right|_{\substack{ x=\cos{\left(t \right)} + 1\\ y=\sin{\left(t \right)} - 2 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \left(x - 1\right)^{2} + \left(y + 2\right)^{2} = 1\ (\text{circle}) \)
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: pass — The first sentence is a generic template but does not specify the actual values (h=1, k=−2, a=b=1) for this problem, making it unclear for a student. The rest of the solution is correct.qwen3.6:27b-mlx: fail (misleading) — The solution fails to explicitly isolate cos(t) and sin(t) from the given parametric equations, which is the core step of 'eliminating the parameter.' It jumps to a general identity and then verifies the result, rather than deriving it. A student would not learn the method of elimination from this.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution fails to explicitly isolate cos(t) and sin(t) from the given parametric equations, which is the core step of 'eliminating the parameter.' It jumps to a general identity and then verifies the result, rather than deriving it. A student would not learn the method of elimination from this.gpt-oss:20b: pass 2026-10-19 — The first sentence is a generic template but does not specify the actual values (h=1, k=−2, a=b=1) for this problem, making it unclear for a student. The rest of the solution is correct.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly isolate cos(t) and sin(t) from the given parametric equations before applying the identity. It jumps straight to the final equation without showing the necessary algebraic steps (x-1 = cos(t), y+2 = sin(t)), making the derivation incomplete and logically disconnected.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence introduces symbols h, k, a, b without defining them or showing that a = b = 1. It does not correctly set up the relationship between x, y and t for the given parametric equations, so a student could be misled about how to eliminate the parameter.qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The setup in step 1 incorrectly states cos t = (x - h)/a, which implies x = h + a cos t. However, the problem gives x = cos(t) + 1, so the correct relation is cos t = x - 1 (i.e., h=1, a=1). The formula provided is generic but applied with undefined variables h, a, b without linking them to the specific problem constants, making the logical jump to the final equation unjustified and confusing. Specifically, it fails to explicitly derive cos(t) = x - 1 and sin(t) = y + 2 from the given parametric equations.gpt-oss:20b: fail (style) 2026-10-18 — [domain objection, downgraded to style] The first sentence introduces variables h, k, a, b that are never defined and does not correctly express the given parametrization. It should state cos t = x−1 and sin t = y+2, then use cos²t+sin²t=1 to eliminate t.
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.