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Eliminating the parameter

Problem 8.367 · medium

Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)} + 1,\ y = 4 \sin{\left(t \right)} + 2 \), and say what kind of curve it is.

cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
\[ \left. \frac{\left(x - 1\right)^{2}}{25} + \frac{\left(y - 2\right)^{2}}{16} - 1 \right|_{\substack{ x=5 \cos{\left(t \right)} + 1\\ y=4 \sin{\left(t \right)} + 2 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \frac{\left(x - 1\right)^{2}}{25} + \frac{\left(y - 2\right)^{2}}{16} = 1\ (\text{ellipse}) \)

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 (error) — The first sentence introduces variables h, k, a, b without defining them and incorrectly states the relationships. It should explicitly use the given parameters: cos t = (x–1)/5 and sin t = (y–2)/4. This misstates the setup and would mislead a student.
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly perform the elimination step by substituting the expressions for cos(t) and sin(t) derived from the parametric equations into the identity. It states a general template and then verifies the final answer, but does not show the derivation required by the prompt to 'eliminate the parameter'.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly perform the elimination step by substituting the expressions for cos(t) and sin(t) derived from the parametric equations into the identity. It states a general template and then verifies the final answer, but does not show the derivation required by the prompt to 'eliminate the parameter'.
gpt-oss:20b: fail (error) 2026-10-18 — The first sentence introduces variables h, k, a, b without defining them and incorrectly states the relationships. It should explicitly use the given parameters: cos t = (x–1)/5 and sin t = (y–2)/4. This misstates the setup and would mislead a student.
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to explicitly eliminate the parameter by deriving the Cartesian equation from the parametric definitions. It merely verifies that the final answer satisfies the parametric equations, which does not demonstrate the elimination process or justify why this is the unique curve.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly identify the parameters h, k, a, and b from the given equations, making the transition to the final formula unjustified. It assumes the reader can map x = 5cos(t) + 1 to the generic form without showing the extraction of center (1, 2) and semi-axes 5 and 4.
gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence introduces symbols h, k, a, b without defining them, and incorrectly states the relationships. It does not correctly set up the elimination process, so a student could be misled about how to derive the ellipse equation.

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.