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Equations of conic sections

Problem 8.306 · easy

Find the equation of the ellipse with vertices \( \displaystyle (\pm 3, 0) \) and foci \( \displaystyle (\pm 2, 0) \).

x²/a² + y²/b² = 1 with a the vertex distance and b² = a² − c².
Reviewed
\[ -4 + 3^{2} = 5 \]
b² = a² − c².✓ Proved
\[ \left. \frac{x^{2}}{9} + \frac{y^{2}}{5} \right|_{\substack{ x=3\\ y=0 }} = 1 \]
The vertex (3, 0) is on the curve.✓ Proved

Answer \( \frac{x^{2}}{9} + \frac{y^{2}}{5} = 1 \)

✓ 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 ellipse, calculates b² using the relationship b² = a² - c², and verifies the vertex. The logic is sound and complete.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the standard form for a horizontal ellipse, calculates b² using the relationship b² = a² - c², and verifies the vertex. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the parameters for a horizontal ellipse and verifies the result. The setup is sound and the algebra is correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameters for a horizontal ellipse and applies the standard relationship between a, b, and c. The setup and calculations are sound.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.