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Equations of conic sections

Problem 8.422 · easy

Find the equation of the ellipse with vertices \( \displaystyle (\pm 2, 0) \) and foci \( \displaystyle (\pm 1, 0) \).

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

Answer \( \frac{x^{2}}{4} + \frac{y^{2}}{3} = 1 \)

Lines: 2 proved, 1 reviewed. 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	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 b^2 using the relationship a^2 = b^2 + c^2, and verifies the vertex. The logic is sound and the result is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the standard form for a horizontal ellipse, calculates b^2 using the relationship a^2 = b^2 + c^2, and verifies the vertex. The logic is sound and the result is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the parameters for a horizontal ellipse and verifies the result.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly justify why the major axis is horizontal (based on the coordinates of the vertices and foci), which is a critical step in selecting the correct standard form. Additionally, step 2 is presented as an unchecked equation rather than a derived value, and step 3 is a verification rather than a derivation of the final equation.
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.