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Equations of conic sections

Problem 8.417 · easy

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 7, 0) \).

x²/a² − y²/b² = 1 with a the vertex distance and b² = c² − a².
Reviewed
\[ -36 + 7^{2} = 13 \]
b² = c² − a².✓ Proved
\[ \left. \frac{x^{2}}{36} - \frac{y^{2}}{13} \right|_{\substack{ x=6\\ y=0 }} = 1 \]
The vertex (6, 0) is on the curve.✓ Proved

Answer \( \frac{x^{2}}{36} - \frac{y^{2}}{13} = 1 \)

✓ 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 hyperbola as horizontal based on the coordinates of the vertices and foci. The calculation of b² is correct, and the final equation matches the derived parameters.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the hyperbola as horizontal based on the coordinates of the vertices and foci. The calculation of b² is correct, and the final equation matches the derived parameters.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the hyperbola as horizontal based on the coordinates of the vertices and foci. The calculation of b² using c² - a² is correct, and the final equation matches the derived parameters.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the hyperbola as horizontal based on the foci and vertices lying on the x-axis. The values for a and c are correctly identified, and the calculation for b² is correct. The final equation matches the stated answer.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.