Equations of conic sections
Problem 8.425 · 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 a single point on the parabola; it does not derive the general equation y²=12x. The second sentence does not establish the equation for all points, so a student could be misled into thinking a single point suffices to prove the parabola’s equation.qwen3.6:27b-mlx: fail (error) — The solution checks a single point to verify the equation, which is insufficient to prove the equation is correct for all points. It fails to derive the equation from the geometric definition or the given focus/directrix parameters.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution checks a single point to verify the equation, which is insufficient to prove the equation is correct for all points. It fails to derive the equation from the geometric definition or the given focus/directrix parameters.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution only verifies a single point on the parabola; it does not derive the general equation y²=12x. The second sentence does not establish the equation for all points, so a student could be misled into thinking a single point suffices to prove the parabola’s equation.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the standard form for a horizontal parabola with vertex at the origin and derives the correct equation. The check with a specific point is valid, though not strictly necessary for the derivation, it does not introduce errors.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — The solution provides a correct definition but fails to derive the equation. It merely verifies a single point, which does not constitute a derivation and teaches the student that checking one point is sufficient to find the equation.gpt-oss:20b: pass 2026-10-15
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.