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Eliminating the parameter

Problem 8.343 · medium

Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} + 2,\ y = \sin{\left(t \right)} \), and say what kind of curve it is.

cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
\[ \left. y^{2} + \frac{\left(x - 2\right)^{2}}{4} - 1 \right|_{\substack{ x=2 \cos{\left(t \right)} + 2\\ y=\sin{\left(t \right)} }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( y^{2} + \frac{\left(x - 2\right)^{2}}{4} = 1\ (\text{ellipse}) \)

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 (style) — [domain objection, downgraded to style] The first sentence introduces a generic ellipse parametrization with symbols h, k, a, b that are never defined or related to the given equations. It does not correctly express the given parametrization (cos t = (x−2)/2, sin t = y).
qwen3.6:27b-mlx: fail (error) — The solution fails to actually eliminate the parameter. It states a generic identity and verifies that the parametric equations satisfy the final result, but it skips the crucial algebraic steps of isolating cos(t) and sin(t) from the given x and y equations to derive the Cartesian form.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to actually eliminate the parameter. It states a generic identity and verifies that the parametric equations satisfy the final result, but it skips the crucial algebraic steps of isolating cos(t) and sin(t) from the given x and y equations to derive the Cartesian form.
gpt-oss:20b: fail (style) 2026-10-18 — [domain objection, downgraded to style] The first sentence introduces a generic ellipse parametrization with symbols h, k, a, b that are never defined or related to the given equations. It does not correctly express the given parametrization (cos t = (x−2)/2, sin t = y).
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to explicitly eliminate the parameter t to derive the Cartesian equation, instead only verifying that the parametric equations satisfy the final result. It also omits the identification of the curve type in the derivation steps, relying solely on the stated answer.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The sentence in step 1 presents a generic formula with undefined variables (h, a, b) without explicitly mapping them to the specific problem parameters (h=2, a=2, b=1). This skips the crucial logical step of identifying the semi-axes and center, making the derivation opaque and potentially confusing for a student trying to learn the method.
gpt-oss:20b: fail (error) 2026-10-14 — The first sentence introduces parameters h, k, a, b without defining them and incorrectly states the relations for cos t and sin t. It does not correctly model the given parametric equations, so the setup is wrong.

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.