Eliminating the parameter
Problem 8.487 · medium
Eliminate the parameter from \( \displaystyle x = 2 \cos{\left(t \right)} + 2,\ y = 2 \sin{\left(t \right)} - 3 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{\left(x - 2\right)^{2}}{4} + \frac{\left(y + 3\right)^{2}}{4} - 1 \right|_{\substack{ x=2 \cos{\left(t \right)} + 2\\ y=2 \sin{\left(t \right)} - 3 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{\left(x - 2\right)^{2}}{4} + \frac{\left(y + 3\right)^{2}}{4} = 1\ (\text{circle}) \)
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: fail (misleading) — The first sentence uses generic symbols h,k,a,b without specifying that here a=2, h=2, k=-3, which could mislead a student about the correct substitutions.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly show the algebraic steps of isolating cos(t) and sin(t) from the given parametric equations, which is the core task of 'eliminating the parameter'. It jumps straight to the identity without demonstrating the substitution, making it an incomplete derivation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to explicitly show the algebraic steps of isolating cos(t) and sin(t) from the given parametric equations, which is the core task of 'eliminating the parameter'. It jumps straight to the identity without demonstrating the substitution, making it an incomplete derivation.gpt-oss:20b: fail (misleading) 2026-10-19 — The first sentence uses generic symbols h,k,a,b without specifying that here a=2, h=2, k=-3, which could mislead a student about the correct substitutions.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly perform the elimination steps (isolating cos t and sin t) required by the problem statement. It jumps from a generic identity to the final equation without showing the substitution of the specific expressions for x and y, making the derivation incomplete.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence is vague and does not correctly identify the specific values of h, k, a, and b for the given parametrization. It also skips the necessary algebraic manipulation to show how the circle equation is obtained, which could mislead a student about the derivation process.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The setup in line 1 incorrectly defines cos t and sin t in terms of generic parameters h, a, b, k without establishing their values from the given equations. Specifically, it fails to isolate cos(t) = (x-2)/2 and sin(t) = (y+3)/2, making the transition to the final equation logically incomplete and confusing.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.