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Eliminating the parameter

Problem 8.494 · medium

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

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

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

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 identify the parameters h, a, and b from the given equations x = 5 cos(t) - 2 and y = 4 sin(t). It states a general formula but does not show the substitution step (e.g., (x+2)/5 = cos(t)) required to derive the final equation, making the logical gap between the setup and the result unjustified.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to explicitly identify the parameters h, a, and b from the given equations x = 5 cos(t) - 2 and y = 4 sin(t). It states a general formula but does not show the substitution step (e.g., (x+2)/5 = cos(t)) required to derive the final equation, making the logical gap between the setup and the result unjustified.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly derive the Cartesian equation from the parametric equations as requested. It states a general identity and then verifies the final answer, but skips the crucial step of isolating cos(t) and sin(t) from x and y to show how the equation is constructed.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly derive the Cartesian equation from the given parametric equations. It states a generic identity and verifies the final answer, but omits the crucial step of isolating cos(t) and sin(t) from x = 5cos(t) - 2 and y = 4sin(t) to show how the specific coefficients and shifts arise.
gpt-oss:20b: pass 2026-10-18

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 2026-10-19 with SymPy 1.14.0.