Home›Calculus 2›Equations of conic sections›Problem 8.297

Equations of conic sections

Problem 8.297 · easy

Find the equation of the ellipse with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 1, 0) \).

x²/a² + y²/b² = 1 with a the vertex distance and b² = a² − c².
Reviewed
\[ -1 + 6^{2} = 35 \]
b² = a² − c².✓ Proved
\[ \left. \frac{x^{2}}{36} + \frac{y^{2}}{35} \right|_{\substack{ x=6\\ y=0 }} = 1 \]
The vertex (6, 0) is on the curve.✓ Proved

Answer \( \frac{x^{2}}{36} + \frac{y^{2}}{35} = 1 \)

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 ellipse parameters and verifies the result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the ellipse parameters and verifies the result.
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.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly identify the center of the ellipse as (0,0) and the orientation as horizontal based on the given vertices and foci. While the final equation is correct, the setup sentence is incomplete and assumes standard position without justification, which is a critical step in modeling the problem.
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.