Eliminating the parameter
Problem 8.225 · medium
Eliminate the parameter from \( \displaystyle x = \sec{\left(t \right)},\ y = \tan{\left(t \right)} \), and say what kind of curve it is.
- sec²t − tan²t = 1.
- \[ \left. x^{2} - y^{2} - 1 \right|_{\substack{ x=\sec{\left(t \right)}\\ y=\tan{\left(t \right)} }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( x^{2} - y^{2} = 1\ (\text{hyperbola}) \)
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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to specify the domain restrictions on x (|x| >= 1), which are necessary because the range of sec(t) does not cover the interval (-1, 1). Without this, the Cartesian equation describes a full hyperbola, while the parametric equations only describe the two branches, making the equivalence incomplete.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution fails to specify the domain restrictions on x (|x| >= 1), which are necessary because the range of sec(t) does not cover the interval (-1, 1). Without this, the Cartesian equation describes a full hyperbola, while the parametric equations only describe the two branches, making the equivalence incomplete.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — The solution fails to identify the curve as a hyperbola, which is explicitly requested. It only verifies that the parametric equations satisfy the algebraic relation, omitting the classification step.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution fails to identify the curve correctly. While x^2 - y^2 = 1 is a hyperbola, the parametric equations x = sec(t) and y = tan(t) only trace the right branch (x >= 1) of the hyperbola, not the entire curve. Stating it is 'a hyperbola' without qualification is misleading because the domain of the parameter excludes the left branch.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.