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Eliminating the parameter

Problem 8.540 · medium

Eliminate the parameter from \( \displaystyle x = \cos{\left(t \right)} - 3,\ y = \sin{\left(t \right)} - 1 \), and say what kind of curve it is.

cos t = (x − h)/a and sin t = (y − k)/b; then cos²t + sin²t = 1.
\[ \left. \left(x + 3\right)^{2} + \left(y + 1\right)^{2} - 1 \right|_{\substack{ x=\cos{\left(t \right)} - 3\\ y=\sin{\left(t \right)} - 1 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \left(x + 3\right)^{2} + \left(y + 1\right)^{2} = 1\ (\text{circle}) \)

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 introduces a generic substitution cos t=(x−h)/a, sin t=(y−k)/b without defining h,k,a,b or relating them to the given parametrization. It does not correctly model the problem and could mislead a student about how to eliminate the parameter.
qwen3.6:27b-mlx: fail (error) — 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 = h + a cos t. However, the given parametric equations are x = cos(t) - 3, meaning h = -3, so the correct relation is cos t = x + 3 (or (x - (-3))/1). The generic formula provided does not match the specific signs in the problem, leading to a confusing and technically incorrect setup description.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-04 — 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 = h + a cos t. However, the given parametric equations are x = cos(t) - 3, meaning h = -3, so the correct relation is cos t = x + 3 (or (x - (-3))/1). The generic formula provided does not match the specific signs in the problem, leading to a confusing and technically incorrect setup description.
gpt-oss:20b: fail (misleading) 2026-11-04 — The first sentence introduces a generic substitution cos t=(x−h)/a, sin t=(y−k)/b without defining h,k,a,b or relating them to the given parametrization. It does not correctly model the problem and could mislead a student about how to eliminate the parameter.
qwen3.6:27b-mlx: fail (error) 2026-11-03 — The setup in step 1 incorrectly defines cos t as (x - h)/a, which implies x = a cos t + h. The given equation is x = cos t - 3, so cos t = x + 3. The formula used leads to the wrong sign for the horizontal shift in the final equation.
gpt-oss:20b: pass 2026-11-03
qwen3.6:27b-mlx: fail (error) 2026-11-03 — The sentence in step 1 incorrectly defines the substitution as cos t = (x - h)/a, whereas the problem requires cos t = x + 3 (i.e., x - (-3)). This implies a sign error in the setup logic, even though the final equation happens to be correct.
gpt-oss:20b: fail (error) 2026-11-03 — The first sentence incorrectly introduces generic parameters h,k,a,b and does not correctly relate the given parametric equations to the circle equation. It misstates the setup and therefore the elimination is not justified.

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 2026-11-04 with SymPy 1.14.0.