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Eliminating the parameter

Problem 8.226 · medium

Eliminate the parameter from \( \displaystyle x = 2 \sec{\left(t \right)},\ y = \tan{\left(t \right)} \), and say what kind of curve it is.

sec²t − tan²t = 1.
\[ \left. \frac{x^{2}}{4} - y^{2} - 1 \right|_{\substack{ x=2 \sec{\left(t \right)}\\ y=\tan{\left(t \right)} }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \frac{x^{2}}{4} - y^{2} = 1\ (\text{hyperbola}) \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 (style) — [domain objection, downgraded to style] The solution derives the implicit equation but fails to address the domain restrictions. The parametric curve is only the right branch of the hyperbola (x >= 2), whereas the equation x^2/4 - y^2 = 1 represents both branches. Identifying the curve solely by the equation without specifying the branch is mathematically imprecise and misleading.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution derives the implicit equation but fails to address the domain restrictions. The parametric curve is only the right branch of the hyperbola (x >= 2), whereas the equation x^2/4 - y^2 = 1 represents both branches. Identifying the curve solely by the equation without specifying the branch is mathematically imprecise and misleading.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution fails to address the domain restrictions. The parametric equations x = 2 sec(t) and y = tan(t) only generate the branches of the hyperbola where |x| >= 2, excluding the region between the vertices. Simply stating the equation is 1/x^2/4 - y^2 = 1 implies the entire hyperbola, which is false for this parameterization.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the trigonometric identity needed to eliminate the parameter and verifies that the parametric equations satisfy the resulting Cartesian equation. The identification of the curve as a hyperbola is correct.
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.