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Equations of conic sections

Problem 8.479 · easy

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 3, 0) \) and foci \( \displaystyle (\pm 6, 0) \).

x²/a² − y²/b² = 1 with a the vertex distance and b² = c² − a².
Reviewed
\[ -9 + 6^{2} = 27 \]
b² = c² − a².✓ Proved
\[ \left. \frac{x^{2}}{9} - \frac{y^{2}}{27} \right|_{\substack{ x=3\\ y=0 }} = 1 \]
The vertex (3, 0) is on the curve.✓ Proved

Answer \( \frac{x^{2}}{9} - \frac{y^{2}}{27} = 1 \)

✓ Nihil obstat Lines: 2 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
3	✓ 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 built from the focal definition (sum or difference of distances, or focus = directrix distance) satisfy the answer

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the standard form for a horizontal hyperbola, calculates b² using the relationship c² = a² + b², and verifies the vertex. The logic is sound and complete.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the standard form for a horizontal hyperbola, calculates b² using the relationship c² = a² + b², and verifies the vertex. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the standard form for a horizontal hyperbola, calculates b² using the relationship c² = a² + b², and verifies the vertex. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the standard form for a horizontal hyperbola, calculates b² using the relationship c² = a² + b², and verifies the vertex. The logic is sound.
gpt-oss:20b: pass 2026-10-16

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/conic_equations, checked 2026-10-19 with SymPy 1.14.0.