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Equations of conic sections

Problem 8.429 · easy

Find the equation of the ellipse with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 3, 0) \).

x²/a² + y²/b² = 1 with a the vertex distance and b² = a² − c².
\[ -9 + 5^{2} = 16 \]
b² = a² − c².✓ Proved
\[ \left. \frac{x^{2}}{25} + \frac{y^{2}}{16} \right|_{\substack{ x=5\\ y=0 }} = 1 \]
The vertex (5, 0) is on the curve.✓ Proved

Answer \( \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1 \)

Lines: 2 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
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: fail (error) — The setup incorrectly defines 'a' as the vertex distance without specifying that 'a' must be the semi-major axis. While the numerical calculation happens to be correct because the vertices are further from the center than the foci, the definition is ambiguous and would fail if the roles of a and b were swapped (e.g., if vertices were on the y-axis). A rigorous solution must explicitly identify the major axis based on the positions of the vertices and foci.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The setup incorrectly defines 'a' as the vertex distance without specifying that 'a' must be the semi-major axis. While the numerical calculation happens to be correct because the vertices are further from the center than the foci, the definition is ambiguous and would fail if the roles of a and b were swapped (e.g., if vertices were on the y-axis). A rigorous solution must explicitly identify the major axis based on the positions of the vertices and foci.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the standard form for a horizontal ellipse, calculates b² using the relationship c² = a² - b², and verifies the result. The setup and logic are sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the standard form for a horizontal ellipse, calculates b^2 using the relationship c^2 = a^2 - b^2, and verifies the vertex. The logic is sound and the final equation is correct.
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.