Equations of conic sections
Problem 8.525 · 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.
- \[ \left. \sqrt{y^{2} + \left(x - 4\right)^{2}} \right|_{\substack{ x=16\\ y=16 }} = \left. \left|{x + 4}\right| \right|_{\substack{ x=16\\ y=16 }} \]The point (16, 16) on it is equally far from focus and directrix.✓ Proved
Answer \( y^{2} = 16 x \)
Lines: 1 proved, 1 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 condition at a single point (16,16) rather than proving it holds for all points on the proposed curve. This gives a false impression that the equation is fully justified.qwen3.6:27b-mlx: fail (error) — The solution checks a single point (16, 16) to verify the equation, which is logically insufficient to derive or prove the equation of the parabola. A single point lying on a curve does not define the curve; the derivation must rely on the general definition for arbitrary (x, y).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-22 — The solution checks a single point (16, 16) to verify the equation, which is logically insufficient to derive or prove the equation of the parabola. A single point lying on a curve does not define the curve; the derivation must rely on the general definition for arbitrary (x, y).gpt-oss:20b: fail (misleading) 2026-10-22 — The solution only verifies the distance condition at a single point (16,16) rather than proving it holds for all points on the proposed curve. This gives a false impression that the equation is fully justified.qwen3.6:27b-mlx: pass 2026-10-21 — 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 valid and consistent with the derived equation.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution fails to derive the equation of the parabola. It states the general form but then merely verifies a single point (16, 16) satisfies the distance property, which does not constitute a derivation of the equation y^2 = 16x.gpt-oss:20b: pass 2026-10-21
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 2026-10-22 with SymPy 1.14.0.