Eliminating the parameter
Problem 8.361 · medium
Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} - 1,\ y = \sin{\left(t \right)} + 3 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \left(x + 1\right)^{2} + \left(y - 3\right)^{2} - 1 \right|_{\substack{ x=\cos{\left(t \right)} - 1\\ y=\sin{\left(t \right)} + 3 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \left(x + 1\right)^{2} + \left(y - 3\right)^{2} = 1\ (\text{ellipse}) \)
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 generic parameters a,b,h,k without specifying that a=b=1, h=−1, k=3, so the reader is misled into thinking a general ellipse is being considered. Moreover, the resulting curve is a circle of radius 1, not an ellipse. The solution therefore contains incorrect and misleading statements.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the curve as an ellipse. Since the coefficients of cos(t) and sin(t) are both 1, the resulting curve is a circle, which is a specific case of an ellipse but should be identified as a circle for precision.
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 cos(t) and sin(t) are both 1, the resulting curve is a circle, which is a specific case of an ellipse but should be identified as a circle for precision.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence introduces generic parameters a,b,h,k without specifying that a=b=1, h=−1, k=3, so the reader is misled into thinking a general ellipse is being considered. Moreover, the resulting curve is a circle of radius 1, not an ellipse. The solution therefore contains incorrect and misleading statements.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution correctly derives the equation of a circle but incorrectly identifies the curve as an ellipse. While a circle is technically a special case of an ellipse, in the context of identifying curves from parametric equations, it is standard and more precise to identify it as a circle. Labeling it merely an 'ellipse' is misleading and suggests a lack of precision in recognizing the specific shape.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution states the resulting curve is an ellipse, but the equation (x+1)²+(y-3)²=1 represents a circle. The generic sentence about cos t and sin t with parameters h,k,a,b is also vague and does not correctly identify the specific values from the given parametrization.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the curve as an ellipse. The equation $(x+1)^2 + (y-3)^2 = 1$ represents a circle, which is a specific case of an ellipse, but in standard calculus contexts, distinguishing between a circle and a general ellipse is important. More critically, the prompt asks to 'say what kind of curve it is', and labeling a circle merely as an 'ellipse' is imprecise and potentially misleading if the distinction is expected. However, the primary error is likely that the system expects 'circle' as the specific answer, or the label 'ellipse' is considered incorrect because the coefficients of $x^2$ and $y^2$ are equal. Let's look closer. A circle IS an ellipse. Is it an error? In many automated grading systems, 'circle' is the expected answer for $x^2+y^2=r^2$. Calling it an ellipse is technically true but often marked wrong in introductory calculus if 'circle' is the intended specific classification. Let's assume the strictness of 'what kind of curve' implies the most specific standard name. Also, the first sentence sets up a generic ellipse formula but the substitution leads to a circle. The final label is the issue.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence introduces variables h, k, a, b without defining them or relating them to the given parametrization. It does not correctly model the problem, so the setup is incorrect.
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.