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Eliminating the parameter

Problem 8.535 · medium

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

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

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

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 perform the elimination step (isolating cos(t) and sin(t) and applying the identity). It jumps from a general template to a verification that the result holds, which does not demonstrate how the equation was derived from the parametric forms.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution fails to explicitly perform the elimination step (isolating cos(t) and sin(t) and applying the identity). It jumps from a general template to a verification that the result holds, which does not demonstrate how the equation was derived from the parametric forms.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: fail (error) 2026-11-03 — The solution fails to explicitly perform the elimination step (solving for cos t and sin t and substituting into the identity). It only verifies that the parametric equations satisfy the final Cartesian equation, which is a check, not a derivation. A student would not learn how to eliminate the parameter from this.
gpt-oss:20b: pass 2026-11-03
qwen3.6:27b-mlx: fail (error) 2026-11-03 — The solution fails to explicitly show the algebraic steps of isolating cos(t) and sin(t) from the given parametric equations before applying the identity. It jumps from a generic template to the final result without demonstrating the substitution x-3 = 4cos(t) and y-3 = 4sin(t), which is the core of the problem.
gpt-oss:20b: fail (style) 2026-11-03 — [domain objection, downgraded to style] The first sentence introduces variables h, k, a, b that are never defined and does not correctly express the given parametrization. It should state that τ satisfies τ = (x-3)/4 and τ = (y-3)/4, leading to the circle equation. The current sentence misrepresents the setup and would mislead a student.

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.