Equations of conic sections
Problem 8.420 · easy
Find the equation of the hyperbola with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 8, 0) \).
- x²/a² − y²/b² = 1 with a the vertex distance and b² = c² − a².Reviewed
- \[ -25 + 8^{2} = 39 \]b² = c² − a².✓ Proved
- \[ \left. \frac{x^{2}}{25} - \frac{y^{2}}{39} \right|_{\substack{ x=5\\ y=0 }} = 1 \]The vertex (5, 0) is on the curve.✓ Proved
Answer \( \frac{x^{2}}{25} - \frac{y^{2}}{39} = 1 \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the hyperbola as horizontal based on the foci and vertices lying on the x-axis. It correctly applies the relationship b² = c² - a² and verifies the vertex satisfies the equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the hyperbola as horizontal based on the foci and vertices lying on the x-axis. It correctly applies the relationship b² = c² - a² and verifies the vertex satisfies the equation.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-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 2026-10-16qwen3.6:27b-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 extracted, 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.