Eliminating the parameter
Problem 8.228 · medium
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} + 1,\ y = 4 \sin{\left(t \right)} - 3 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \left(x - 1\right)^{2} + \frac{\left(y + 3\right)^{2}}{16} - 1 \right|_{\substack{ x=\cos{\left(t \right)} + 1\\ y=4 \sin{\left(t \right)} - 3 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \left(x - 1\right)^{2} + \frac{\left(y + 3\right)^{2}}{16} = 1\ (\text{ellipse}) \)
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 (misleading) — The first sentence introduces generic parameters h,k,a,b without relating them to the given equations, so a student might not correctly identify that a=1, b=4, h=1, k=−3. The subsequent algebraic step is correct, but the setup is not clearly tied to the problem statement.qwen3.6:27b-mlx: fail (error) — The solution fails to actually perform the elimination of the parameter. It states a general identity and verifies that the final answer is correct, but it does not show the derivation steps (isolating cos(t) and sin(t) from the given x and y equations) required to 'eliminate the parameter'.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to actually perform the elimination of the parameter. It states a general identity and verifies that the final answer is correct, but it does not show the derivation steps (isolating cos(t) and sin(t) from the given x and y equations) required to 'eliminate the parameter'.gpt-oss:20b: fail (misleading) 2026-10-17 — The first sentence introduces generic parameters h,k,a,b without relating them to the given equations, so a student might not correctly identify that a=1, b=4, h=1, k=−3. The subsequent algebraic step is correct, but the setup is not clearly tied to the problem statement.qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The sentence in step 1 presents a generic template with undefined variables (h, a, b, k) but fails to explicitly state the values derived from the problem statement (h=1, k=-3, a=1, b=4). This leaves the logical gap between the general identity and the specific solution unbridged, potentially confusing a student about how the specific constants were chosen.gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence introduces generic parameters h,k,a,b without relating them to the specific problem, so a student would incorrectly think any ellipse can be obtained by arbitrary shifts and scalings. It should explicitly state cos t = x−1 and sin t = (y+3)/4 before applying the Pythagorean identity.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to actually perform the elimination steps. It states a generic identity and then verifies the final answer, but does not show the derivation of the Cartesian equation from the parametric ones as requested.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.