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Eliminating the parameter

Problem 8.233 · medium

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

sec²t − tan²t = 1.
\[ \left. \frac{x^{2}}{9} - y^{2} - 1 \right|_{\substack{ x=3 \sec{\left(t \right)}\\ y=\tan{\left(t \right)} }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \frac{x^{2}}{9} - y^{2} = 1\ (\text{hyperbola}) \)

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: pass
qwen3.6:27b-mlx: fail (misleading) — The solution fails to identify the curve as a hyperbola, which is required by the problem statement. It only verifies that the parametric equations satisfy the algebraic relation, but does not conclude the geometric classification.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution fails to identify the curve as a hyperbola, which is required by the problem statement. It only verifies that the parametric equations satisfy the algebraic relation, but does not conclude the geometric classification.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — The solution fails to identify the curve as a hyperbola, which is required by the problem statement. It only verifies that the parametric equations satisfy the Cartesian equation, but does not state the geometric classification.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution fails to identify the domain restrictions implied by the parameter t. Since x = 3 sec(t), x cannot be in (-3, 3), meaning the curve is only the two branches of the hyperbola, not the entire hyperbola defined by x^2/9 - y^2 = 1.
gpt-oss:20b: pass 2026-10-14

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.