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Eliminating the parameter

Problem 8.341 · medium

Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} - 1,\ y = 2 \sin{\left(t \right)} - 3 \), and say what kind of curve it is.

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

Answer \( \frac{\left(x + 1\right)^{2}}{4} + \frac{\left(y + 3\right)^{2}}{4} = 1\ (\text{ellipse}) \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the curve as an ellipse. Since the coefficients of the squared terms are equal (both denominators are 4), the curve is a circle. While a circle is technically a special case of an ellipse, in the context of identifying curves from parametric equations, distinguishing between a circle and a non-circular ellipse is standard and expected. Labeling it merely an 'ellipse' is misleading and imprecise.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the curve as an ellipse. Since the coefficients of the squared terms are equal (both denominators are 4), the curve is a circle. While a circle is technically a special case of an ellipse, in the context of identifying curves from parametric equations, distinguishing between a circle and a non-circular ellipse is standard and expected. Labeling it merely an 'ellipse' is misleading and imprecise.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly identifies the curve as an ellipse. Since the coefficients of the squared terms are equal (both denominators are 4), the curve is a circle. While a circle is technically a special case of an ellipse, in calculus contexts, distinguishing between the two is standard; labeling it merely an 'ellipse' is imprecise and often considered incorrect when 'circle' is the specific answer.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the curve as an ellipse. Since the coefficients of cos(t) and sin(t) are equal (both are 2), the curve is a circle, which is a specific case of an ellipse but should be identified as such for precision. The algebraic form is correct, but the classification is imprecise/misleading in a calculus context where distinguishing circles from general ellipses is standard.
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.