Equations of conic sections
Problem 8.527 · easy
Find the equation of the parabola with focus \( \displaystyle (0, 2) \) and directrix \( \displaystyle y = -2 \).
- 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.
- \[ \left. \sqrt{x^{2} + \left(y - 2\right)^{2}} \right|_{\substack{ x=8\\ y=8 }} = \left. \left|{y + 2}\right| \right|_{\substack{ x=8\\ y=8 }} \]The point (8, 8) on it is equally far from focus and directrix.✓ Proved
Answer \( x^{2} = 8 y \)
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 derive the parabola’s equation; it merely verifies a single point. The setup is correct, but the reasoning is incomplete and does not show how x²=8y follows from the focus and directrix.qwen3.6:27b-mlx: fail (error) — The solution fails to derive the equation of the parabola. It states the general form and verifies a single point, but does not perform the derivation (e.g., using the definition of a parabola or the standard formula) to arrive at x^2 = 8y.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to derive the equation of the parabola. It states the general form and verifies a single point, but does not perform the derivation (e.g., using the definition of a parabola or the standard formula) to arrive at x^2 = 8y.gpt-oss:20b: fail (error) 2026-10-25 — The solution does not derive the parabola’s equation; it merely verifies a single point. The setup is correct, but the reasoning is incomplete and does not show how x²=8y follows from the focus and directrix.qwen3.6:27b-mlx: fail (error) 2026-10-24 — The solution fails to derive the equation of the parabola. It states the general form and verifies a single point, but does not perform the necessary algebraic steps to determine the parameter p or arrive at the final equation x^2 = 8y.gpt-oss:20b: pass 2026-10-24qwen3.6:27b-mlx: fail (error) 2026-10-24 — The solution fails to derive the equation of the parabola. It states the definition and then verifies that a single point (8, 8) satisfies the condition, but it does not perform the algebraic derivation to arrive at the general equation x^2 = 8y.gpt-oss:20b: fail (misleading) 2026-10-24 — The first sentence incorrectly states the standard forms: it claims both y²=4px and x²=4py, but for a vertical parabola the correct form is x²=4py. It also omits the key step that p equals the distance from the vertex to the focus (2).
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 2026-10-25 with SymPy 1.14.0.