Equations of conic sections
Problem 8.430 · easy
Find the equation of the parabola with focus \( \displaystyle (2, 0) \) and directrix \( \displaystyle x = -2 \).
- 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.
- \[ \left. \sqrt{y^{2} + \left(x - 2\right)^{2}} \right|_{\substack{ x=8\\ y=8 }} = \left. \left|{x + 2}\right| \right|_{\substack{ x=8\\ y=8 }} \]The point (8, 8) on it is equally far from focus and directrix.✓ Proved
Answer \( y^{2} = 8 x \)
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 (error) — The solution does not correctly derive the parabola’s equation. It merely checks that a single point (8,8) satisfies the distance condition but never uses the focus–directrix definition to find the parameter p or to establish y²=8x. The setup is correct, but the reasoning is incomplete and misleading.qwen3.6:27b-mlx: fail (misleading) — The solution relies on a single test point to verify the equation, which is insufficient to derive or prove the general equation of the parabola. It fails to demonstrate the derivation from the definition for an arbitrary point (x, y).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution relies on a single test point to verify the equation, which is insufficient to derive or prove the general equation of the parabola. It fails to demonstrate the derivation from the definition for an arbitrary point (x, y).gpt-oss:20b: fail (error) 2026-10-18 — The solution does not correctly derive the parabola’s equation. It merely checks that a single point (8,8) satisfies the distance condition but never uses the focus–directrix definition to find the parameter p or to establish y²=8x. The setup is correct, but the reasoning is incomplete and misleading.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 verifies the result with a valid point.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution only checks that a single point (8,8) satisfies the distance condition; it does not derive the general equation y²=8x. The reasoning is incomplete and could mislead a student into thinking that verifying one point proves the parabola’s equation.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the geometric definition and the standard form for a horizontal parabola with vertex at the origin. The check with point (8,8) is valid as it satisfies y^2 = 8x.gpt-oss:20b: fail (misleading) 2026-10-15 — The solution only checks a single point (8,8) to assert the parabola’s equation, which is insufficient and could mislead a student into thinking one point determines the entire conic. The derivation does not actually produce the general equation y²=8x.
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.