Eliminating the parameter
Problem 8.372 · medium
Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)} + 1,\ y = \sin{\left(t \right)} - 1 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{\left(x - 1\right)^{2}}{25} + \left(y + 1\right)^{2} - 1 \right|_{\substack{ x=5 \cos{\left(t \right)} + 1\\ y=\sin{\left(t \right)} - 1 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{\left(x - 1\right)^{2}}{25} + \left(y + 1\right)^{2} = 1\ (\text{ellipse}) \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 introduces symbols h, k, a, b without defining them, and incorrectly states the general form of the parametrization. It does not correctly relate x and y to the given expressions, so a student could be misled about how to eliminate the parameter.qwen3.6:27b-mlx: fail (error) — The solution fails to actually eliminate the parameter. It states a general identity and verifies that the parametric equations satisfy the final Cartesian equation, but it does not show the derivation steps (isolating cos(t) and sin(t) from the given x and y equations) required to 'eliminate the parameter'.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to actually eliminate the parameter. It states a general identity and verifies that the parametric equations satisfy the final Cartesian equation, but it does not show the derivation steps (isolating cos(t) and sin(t) from the given x and y equations) required to 'eliminate the parameter'.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence introduces symbols h, k, a, b without defining them, and incorrectly states the general form of the parametrization. It does not correctly relate x and y to the given expressions, so a student could be misled about how to eliminate the parameter.qwen3.6:27b-mlx: fail (misleading) 2026-10-16 — The solution identifies the curve as an ellipse, but it is actually a circle (a special case of an ellipse). While mathematically an ellipse, calling it an ellipse without noting it is a circle is misleading in this context where the coefficients are distinct squares of radii (5 and 1), implying a general ellipse, but the specific form (x-1)^2/25 + (y+1)^2/1 = 1 is indeed an ellipse. Wait, a circle is an ellipse. Is it misleading? Usually, if a=5 and b=1, it's an ellipse. If a=b, it's a circle. Here a=5, b=1. So it is an ellipse. The statement is correct. Let me re-read carefully. x = 5cos(t)+1, y=sin(t)-1. (x-1)/5 = cos(t), (y+1)/1 = sin(t). Sum of squares is 1. This is an ellipse with semi-major axis 5 and semi-minor axis 1. So 'ellipse' is the correct classification. Why did I think it was a circle? I misread the coefficients. 5 vs 1. Okay, so the classification is correct. Is there any other issue? The sentence says 'cos t = (x-h)/a ...'. It doesn't explicitly define h, k, a, b from the given equations, but the next line shows the substitution. The logic holds. The verdict should be pass.gpt-oss:20b: fail (misleading) 2026-10-16 — The first sentence introduces variables h, k, a, b without defining them and incorrectly states the relationships. It should explicitly write cos t=(x−1)/5 and sin t=y+1, then use the identity cos²t+sin²t=1. As written, a student could be misled about the correct substitution.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly show the algebraic steps of isolating cos(t) and sin(t) from the parametric equations before applying the identity. It jumps from a general template to the final result without demonstrating the specific substitution for this problem, which is the core of the task 'Eliminate the parameter'.gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence introduces symbols h, k, a, b without defining them and incorrectly states the relationships between x, y and cos t, sin t. It misleads a student about how to eliminate the parameter.
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.