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Equations of conic sections

Problem 8.332 · easy

Find the equation of the parabola with focus \( \displaystyle (-4, 0) \) and directrix \( \displaystyle x = 4 \).

A parabola is the set of points equally far from the focus and the directrix; with vertex at the origin that gives y² = 4px (or x² = 4py), p the focus's distance from the vertex.
Reviewed
\[ \left. \sqrt{y^{2} + \left(x + 4\right)^{2}} \right|_{\substack{ x=-16\\ y=16 }} = \left. \left|{x - 4}\right| \right|_{\substack{ x=-16\\ y=16 }} \]
The point (-16, 16) on it is equally far from focus and directrix.✓ Proved

Answer \( y^{2} = - 16 x \)

Lines: 1 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
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 geometric definition and the resulting standard form equation. The verification step confirms a point on the curve satisfies the definition.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the geometric definition and the resulting standard form equation. The verification step confirms a point on the curve satisfies the definition.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the geometric definition and the standard form for a horizontal parabola. The algebraic check in line 2 is valid, though the derivation of the final equation is implicit rather than explicit, the stated answer is correct and consistent with the setup.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to derive the equation of the parabola. It states the definition and verifies a single point, but does not perform the algebraic derivation (e.g., using the distance formula for a general point (x,y)) required to find the equation y^2 = -16x.
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 2026-10-18 with SymPy 1.14.0.