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Equations of conic sections

Problem 8.307 · easy

Find the equation of the parabola with focus \( \displaystyle (1, 0) \) and directrix \( \displaystyle x = -1 \).

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
\[ \left. \sqrt{y^{2} + \left(x - 1\right)^{2}} \right|_{\substack{ x=4\\ y=4 }} = \left. \left|{x + 1}\right| \right|_{\substack{ x=4\\ y=4 }} \]
The point (4, 4) on it is equally far from focus and directrix.✓ Proved

Answer \( y^{2} = 4 x \)

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 standard form for a horizontal parabola with vertex at the origin. The check with a specific point is consistent with the derived equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the geometric definition and the standard form for a horizontal parabola with vertex at the origin. The check with a specific point is consistent with the derived equation.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to derive the equation of the parabola. It states the definition and verifies that a single point (4, 4) lies on the curve, but it does not perform the general algebraic derivation required to find the equation y^2 = 4x.
gpt-oss:20b: fail (misleading) 2026-10-15 — The solution claims that the point (4,4) lies on the parabola and substitutes it, but it never actually determines the focal length p or shows that the resulting equation is y²=4x. The reasoning is incomplete and could mislead a student into thinking that a single arbitrary point suffices to determine the parabola’s equation.
qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution relies on checking a single point to validate the equation, which is logically insufficient to derive or prove the general equation of a parabola. A correct derivation requires equating the distance formulas for a general point (x, y) and simplifying.
gpt-oss:20b: fail (misleading) 2026-10-14 — The solution only checks one point (4,4) for equality of distances; this does not establish the equation of the parabola. The method is incomplete and could mislead a student into thinking a single point suffices to determine the parabola.

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.