Eliminating the parameter
Problem 8.492 · medium
Eliminate the parameter from \( \displaystyle x = 5 \cos{\left(t \right)},\ y = 5 \sin{\left(t \right)} + 3 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{x^{2}}{25} + \frac{\left(y - 3\right)^{2}}{25} - 1 \right|_{\substack{ x=5 \cos{\left(t \right)}\\ y=5 \sin{\left(t \right)} + 3 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{x^{2}}{25} + \frac{\left(y - 3\right)^{2}}{25} = 1\ (\text{ellipse}) \)
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: passqwen3.6:27b-mlx: fail (misleading) — The solution identifies the curve as an ellipse, but it is actually a circle. While a circle is a special case of an ellipse, labeling it merely as an ellipse is misleading in this context where the specific shape (equal semi-axes) is evident and standard terminology distinguishes circles from non-circular ellipses.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution identifies the curve as an ellipse, but it is actually a circle. While a circle is a special case of an ellipse, labeling it merely as an ellipse is misleading in this context where the specific shape (equal semi-axes) is evident and standard terminology distinguishes circles from non-circular ellipses.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution identifies the curve as an ellipse, but the equation x^2/25 + (y-3)^2/25 = 1 represents a circle (a specific case of an ellipse). While technically true that a circle is an ellipse, in introductory calculus contexts, distinguishing between circles and general ellipses is important for precision. Labeling it merely an 'ellipse' without noting it is a circle is misleadingly imprecise.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence introduces parameters h, k, a, b without defining them, and incorrectly states that the resulting curve is an ellipse. In fact, with a=5, b=5, h=0, k=3 the curve is a circle of radius 5 centered at (0,3).qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the curve as an ellipse. Since the denominators are equal (both 25), the curve is a circle. While a circle is technically a special case of an ellipse, in standard calculus contexts, distinguishing them is required for correctness.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.