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Eliminating the parameter

Problem 8.231 · medium

Eliminate the parameter from \( \displaystyle x = 3 \cos{\left(t \right)} - 3,\ y = 4 \sin{\left(t \right)} \), and say what kind of curve it is.

cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
\[ \left. \frac{y^{2}}{16} + \frac{\left(x + 3\right)^{2}}{9} - 1 \right|_{\substack{ x=3 \cos{\left(t \right)} - 3\\ y=4 \sin{\left(t \right)} }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \frac{y^{2}}{16} + \frac{\left(x + 3\right)^{2}}{9} = 1\ (\text{ellipse}) \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly solve for cos(t) and sin(t) in terms of x and y, which is the core of 'eliminating the parameter'. It states a generic identity and then verifies the final answer, skipping the derivation steps required by the problem statement.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to explicitly solve for cos(t) and sin(t) in terms of x and y, which is the core of 'eliminating the parameter'. It states a generic identity and then verifies the final answer, skipping the derivation steps required by the problem statement.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly derive the Cartesian equation from the parametric definitions. It states a general identity and verifies the final answer, but skips the crucial step of isolating cos(t) and sin(t) from the given x and y equations to show how the specific constants (h=-3, a=3, b=4) are obtained.
gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence introduces a generic relation cos t=(x−h)/a, sin t=(y−k)/b without specifying the actual values h, k, a, b for the given parametrization. This omission makes the step unclear and could mislead a student into thinking the formula applies without further identification of the constants.
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly derive the Cartesian equation from the parametric equations, instead stating a generic identity and then verifying the final answer. It does not show the necessary steps of isolating cos(t) and sin(t) and applying the Pythagorean identity, which is the core of the problem.
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.