Equations of conic sections
Problem 8.334 · easy
Find the equation of the parabola with focus \( \displaystyle (-3, 0) \) and directrix \( \displaystyle x = 3 \).
- 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.
- \[ \left. \sqrt{y^{2} + \left(x + 3\right)^{2}} \right|_{\substack{ x=-12\\ y=12 }} = \left. \left|{x - 3}\right| \right|_{\substack{ x=-12\\ y=12 }} \]The point (-12, 12) on it is equally far from focus and directrix.✓ Proved
Answer \( y^{2} = - 12 x \)
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: fail (misleading) — The solution only verifies the distance equality at a single point (-12,12); it does not derive the parabola’s equation from the focus-directrix definition. A student could incorrectly conclude that checking one point is sufficient to determine the entire curve.qwen3.6:27b-mlx: fail (error) — The solution fails to derive the equation of the parabola. It merely verifies that a single point satisfies the definition, which does not constitute finding the equation. The stated answer is correct, but the provided steps do not lead to it.
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 merely verifies that a single point satisfies the definition, which does not constitute finding the equation. The stated answer is correct, but the provided steps do not lead to it.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution only verifies the distance equality at a single point (-12,12); it does not derive the parabola’s equation from the focus-directrix definition. A student could incorrectly conclude that checking one point is sufficient to determine the entire curve.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the standard form for a horizontal parabola with vertex at the origin and verifies a point on the curve. The stated answer y^2 = -12x is correct for the given focus and directrix.gpt-oss:20b: fail (misleading) 2026-10-15 — The solution only verifies the distance equality at a single point; it does not derive the parabola’s equation from the definition, so a student could incorrectly conclude that checking one point suffices to determine the equation.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to derive the equation of the parabola. It states the definition and checks a single point, but never performs the algebraic derivation (setting distance to focus equal to distance to directrix for a general point (x,y)) required to find the equation y^2 = -12x.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.