Eliminating the parameter
Problem 8.536 · medium
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} - 3,\ y = 4 \sin{\left(t \right)} + 2 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{\left(x + 3\right)^{2}}{4} + \frac{\left(y - 2\right)^{2}}{16} - 1 \right|_{\substack{ x=2 \cos{\left(t \right)} - 3\\ y=4 \sin{\left(t \right)} + 2 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{\left(x + 3\right)^{2}}{4} + \frac{\left(y - 2\right)^{2}}{16} = 1\ (\text{ellipse}) \)
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 symbols h, k, a, b without defining them, and incorrectly writes cos t = (x−h)/a and sin t = (y−k)/b. The correct substitutions are cos t = (x+3)/2 and sin t = (y−2)/4. This vague statement could mislead a student about how to eliminate the parameter.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution states a generic formula with undefined variables h, a, and b, but fails to explicitly identify their values from the given parametric equations (h=-3, a=2, k=2, b=4). This skips the crucial step of matching the specific problem to the general identity, making the derivation opaque and potentially confusing for a student.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-11-04 — [domain objection, downgraded to style] The solution states a generic formula with undefined variables h, a, and b, but fails to explicitly identify their values from the given parametric equations (h=-3, a=2, k=2, b=4). This skips the crucial step of matching the specific problem to the general identity, making the derivation opaque and potentially confusing for a student.gpt-oss:20b: fail (misleading) 2026-11-04 — The first sentence introduces symbols h, k, a, b without defining them, and incorrectly writes cos t = (x−h)/a and sin t = (y−k)/b. The correct substitutions are cos t = (x+3)/2 and sin t = (y−2)/4. This vague statement could mislead a student about how to eliminate the parameter.qwen3.6:27b-mlx: fail (misleading) 2026-11-03 — The solution identifies the curve as an ellipse but fails to specify the direction of traversal (clockwise vs. counter-clockwise) or the range of the curve, which are standard parts of 'saying what kind of curve it is' in parametric contexts. More critically, the sentence in step 1 is a heuristic rather than a rigorous derivation; it assumes the form of the ellipse without showing how h, k, a, and b are extracted from the given parametric equations, which could mislead a student into guessing coefficients rather than solving for them.gpt-oss:20b: pass 2026-11-03qwen3.6:27b-mlx: fail (error) 2026-11-03 — The solution fails to explicitly derive the Cartesian equation from the parametric definitions, instead stating a generic identity and then verifying the result. It does not show the necessary algebraic steps (isolating cos(t) and sin(t)) required to 'eliminate the parameter' as requested.gpt-oss:20b: pass 2026-11-03
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 2026-11-04 with SymPy 1.14.0.