Eliminating the parameter
Problem 8.340 · medium
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} - 2,\ y = 4 \sin{\left(t \right)} - 3 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \left(x + 2\right)^{2} + \frac{\left(y + 3\right)^{2}}{16} - 1 \right|_{\substack{ x=\cos{\left(t \right)} - 2\\ y=4 \sin{\left(t \right)} - 3 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \left(x + 2\right)^{2} + \frac{\left(y + 3\right)^{2}}{16} = 1\ (\text{ellipse}) \)
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 (error) — The first sentence introduces symbols h, k, a, b without defining them and incorrectly relates them to x and y. It does not correctly express the given parametric equations, so the setup is wrong.qwen3.6:27b-mlx: fail (misleading) — The solution verifies that the parametric equations satisfy the Cartesian equation, but it fails to demonstrate that the Cartesian equation is derived from the parametric ones (i.e., it does not show the elimination of the parameter). This is a verification, not a derivation, and does not teach the student how to eliminate the parameter.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution verifies that the parametric equations satisfy the Cartesian equation, but it fails to demonstrate that the Cartesian equation is derived from the parametric ones (i.e., it does not show the elimination of the parameter). This is a verification, not a derivation, and does not teach the student how to eliminate the parameter.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence introduces symbols h, k, a, b without defining them and incorrectly relates them to x and y. It does not correctly express the given parametric equations, so the setup is wrong.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to actually eliminate the parameter. It states a generic identity and verifies that the parametric equations satisfy the final Cartesian equation, but it does not show the derivation (isolating cos(t) and sin(t) from x and y) required to 'eliminate the parameter'.gpt-oss:20b: fail (error) 2026-10-16 — The first sentence introduces symbols h, k, a, b without defining them, and incorrectly states the general form. It does not correctly model the given parametric equations, so the setup is wrong.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the curve as an ellipse and provides the correct Cartesian equation. The reasoning, while brief, accurately reflects the standard method of eliminating the parameter using the Pythagorean identity.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 2026-10-18 with SymPy 1.14.0.