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Eliminating the parameter

Problem 8.533 · medium

Eliminate the parameter from \( \displaystyle x = 2 \sec{\left(t \right)},\ y = 3 \tan{\left(t \right)} \), and say what kind of curve it is.

sec²t − tan²t = 1.
\[ \left. \frac{x^{2}}{4} - \frac{y^{2}}{9} - 1 \right|_{\substack{ x=2 \sec{\left(t \right)}\\ y=3 \tan{\left(t \right)} }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( \frac{x^{2}}{4} - \frac{y^{2}}{9} = 1\ (\text{hyperbola}) \)

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 (error) — 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 classify the curve.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-04 — 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 classify the curve.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: fail (style) 2026-11-03 — [domain objection, downgraded to style] The solution fails to specify the domain of the parameter t, which is crucial for determining the exact curve. Since sec(t) is undefined at t = pi/2 + k*pi, the parametric equations do not cover the entire hyperbola (specifically, the points where x is between -2 and 2 are missing, and the vertices are approached but the gap between branches is not part of the image). More importantly, the question asks to 'say what kind of curve it is', and simply stating 'hyperbola' without noting that it is a specific branch or the whole hyperbola minus the segment between vertices is incomplete. However, the primary error is that the algebraic elimination produces the full hyperbola, but the parametric form only traces parts of it (specifically, it traces the right branch for t in (-pi/2, pi/2) and the left branch for t in (pi/2, 3pi/2), but misses the points where cos(t)=0). Actually, sec(t) ranges from (-inf, -1] U [1, inf), so x ranges from (-inf, -2] U [2, inf). The equation x^2/4 - y^2/9 = 1 describes the entire hyperbola. The parametric equations describe the entire hyperbola except for the points where the tangent is vertical? No, tan(t) goes to infinity as sec(t) goes to infinity. The issue is that for any point on the hyperbola, there exists a t such that x=2sec(t) and y=3tan(t)? Yes, because for any x with |x|>=2, we can find t such that sec(t)=x/2, and then tan(t) is determined up to sign, but y determines the sign. So the image is the entire hyperbola. Wait. Is the image the entire hyperbola? Yes. But the solution does not explicitly state that the curve is a hyperbola, it just gives the equation. The prompt asks to 'say what kind of curve it is'. The stated answer does say '(hyperbola)'. The solution steps are: 1. Identity. 2. Verification. It does not explicitly conclude 'Therefore, the curve is a hyperbola'. It just shows the point satisfies the equation. This is a gap in logic. The student needs to connect 'satisfies equation of hyperbola' to 'is a hyperbola'. Also, strictly speaking, one should check if the mapping is surjective onto the hyperbola. It is. But the step from 'satisfies equation' to 'is the curve' is missing. The solution is incomplete.
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.