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Eliminating the parameter

Problem 8.539 · medium

Eliminate the parameter from \( \displaystyle x = 4 \sec{\left(t \right)},\ y = 2 \tan{\left(t \right)} \), and say what kind of curve it is.

sec²t − tan²t = 1.
Reviewed
\[ \left. \frac{x^{2}}{16} - \frac{y^{2}}{4} - 1 \right|_{\substack{ x=4 \sec{\left(t \right)}\\ y=2 \tan{\left(t \right)} }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \frac{x^{2}}{16} - \frac{y^{2}}{4} = 1\ (\text{hyperbola}) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-04 — 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 2026-11-04
qwen3.6:27b-mlx: pass 2026-11-03 — The solution correctly identifies the trigonometric identity needed to eliminate the parameter and verifies that the resulting Cartesian equation is satisfied by the parametric equations. The identification of the curve as a hyperbola is correct.
gpt-oss:20b: pass 2026-11-03
qwen3.6:27b-mlx: pass 2026-11-03 — The solution correctly identifies the trigonometric identity needed to eliminate the parameter and verifies that the resulting Cartesian equation is satisfied by the parametric equations. The identification of the curve as a hyperbola is correct.
gpt-oss:20b: pass 2026-11-03

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.