Eliminating the parameter
Problem 8.229 · medium
Eliminate the parameter from \( \displaystyle x = 4 \cos{\left(t \right)} - 3,\ y = 4 \sin{\left(t \right)} + 1 \), and say what kind of curve it is.
- cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
- \[ \left. \frac{\left(x + 3\right)^{2}}{16} + \frac{\left(y - 1\right)^{2}}{16} - 1 \right|_{\substack{ x=4 \cos{\left(t \right)} - 3\\ y=4 \sin{\left(t \right)} + 1 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( \frac{\left(x + 3\right)^{2}}{16} + \frac{\left(y - 1\right)^{2}}{16} = 1\ (\text{circle}) \)
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 symbols h, k, a, b without relating them to the given parametrization. It does not correctly state that cos t=(x+3)/4 and sin t=(y−1)/4, so a student could be misled about how the elimination is performed.qwen3.6:27b-mlx: fail (misleading) — The sentence in step 1 incorrectly states the substitution formulas as cos t = (x - h)/a and sin t = (y - k)/b, which implies x = a cos t + h. However, the problem has x = 4 cos t - 3, meaning h = -3, so the correct relation is cos t = (x + 3)/4. Using the generic 'h' without specifying its value or sign relative to the given equation is confusing and technically incorrect as a direct substitution rule for this specific problem.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The sentence in step 1 incorrectly states the substitution formulas as cos t = (x - h)/a and sin t = (y - k)/b, which implies x = a cos t + h. However, the problem has x = 4 cos t - 3, meaning h = -3, so the correct relation is cos t = (x + 3)/4. Using the generic 'h' without specifying its value or sign relative to the given equation is confusing and technically incorrect as a direct substitution rule for this specific problem.gpt-oss:20b: fail (misleading) 2026-10-17 — The first sentence introduces generic symbols h, k, a, b without relating them to the given parametrization. It does not correctly state that cos t=(x+3)/4 and sin t=(y−1)/4, so a student could be misled about how the elimination is performed.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly perform the algebraic elimination steps (isolating cos(t) and sin(t) and substituting them into the identity). It merely states a generic template and verifies the final result, which does not demonstrate the method of elimination requested by the problem.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence states a generic relationship “cos t = (x−h)/a and sin t = (y−k)/b” without specifying the actual values h, k, a, b for this problem. It does not correctly model the given parametric equations, which require (x+3)/4 = cos t and (y−1)/4 = sin t. Consequently the setup is incorrect and the subsequent conclusion is not justified.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly show the algebraic steps of isolating cos(t) and sin(t) from the given parametric equations before applying the identity. It jumps from a generic formula to the final result without demonstrating the substitution x+3 = 4cos(t) and y-1 = 4sin(t), which is the core of the problem.gpt-oss:20b: fail (misleading) 2026-10-14 — The first sentence is too vague: it does not specify the actual values of h, k, a, and b for the given parametrization. A student could incorrectly assume any values, leading to confusion 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 2026-10-17 with SymPy 1.14.0.