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Equations of conic sections

Problem 8.423 · 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.
\[ \left. \sqrt{y^{2} + \left(x + 1\right)^{2}} \right|_{\substack{ x=-4\\ y=4 }} = \left. \left|{x - 1}\right| \right|_{\substack{ x=-4\\ y=4 }} \]
The point (-4, 4) on it is equally far from focus and directrix.✓ Proved

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

Lines: 1 proved, 1 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
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 solution fails to derive the equation of the parabola. It states the definition and verifies a single point, but never performs the algebraic derivation (setting distance to focus equal to distance to directrix) to arrive at the stated answer y^2 = -4x.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to derive the equation of the parabola. It states the definition and verifies a single point, but never performs the algebraic derivation (setting distance to focus equal to distance to directrix) to arrive at the stated answer y^2 = -4x.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to derive the equation of the parabola. It states a general form and verifies a single point, but does not perform the necessary algebraic steps to arrive at the final equation y^2 = -4x.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to derive the equation of the parabola. It states the definition and checks a single point, but does not perform the algebraic derivation (setting distance to focus equal to distance to directrix) required to find the general equation y^2 = -4x.
gpt-oss:20b: fail (misleading) 2026-10-15 — The sentence claims that with vertex at the origin a parabola has equation y²=4px, but this ignores the sign of p. For a focus left of the vertex, p is negative and the correct equation is y²=−4x. The statement would mislead a student into thinking the sign of p is irrelevant.

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.