Equations of conic sections
Problem 8.431 · easy
Find the equation of the parabola with focus \( \displaystyle (0, -1) \) and directrix \( \displaystyle y = 1 \).
- 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.
- \[ \left. \sqrt{x^{2} + \left(y + 1\right)^{2}} \right|_{\substack{ x=4\\ y=-4 }} = \left. \left|{y - 1}\right| \right|_{\substack{ x=4\\ y=-4 }} \]The point (4, -4) on it is equally far from focus and directrix.✓ Proved
Answer \( x^{2} = - 4 y \)
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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly assumes the vertex is at the origin without justification. The vertex is actually at (0, 0) only if the focus and directrix are symmetric about the origin, which they are, but the derivation skips the crucial step of identifying the vertex location relative to the focus and directrix to determine the sign and value of p correctly. More importantly, the equation y^2 = 4px is for a horizontal parabola, but the directrix y=1 implies a vertical parabola, so the form should be x^2 = 4py. The solution mentions x^2 = 4py but then uses a test point that doesn't help derive the equation generally, and the logic is disjointed.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly assumes the vertex is at the origin without justification. The vertex is actually at (0, 0) only if the focus and directrix are symmetric about the origin, which they are, but the derivation skips the crucial step of identifying the vertex location relative to the focus and directrix to determine the sign and value of p correctly. More importantly, the equation y^2 = 4px is for a horizontal parabola, but the directrix y=1 implies a vertical parabola, so the form should be x^2 = 4py. The solution mentions x^2 = 4py but then uses a test point that doesn't help derive the equation generally, and the logic is disjointed.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The setup incorrectly assumes the vertex is at the origin without justification, although it happens to be true here. More critically, the solution does not derive the general equation from the definition; it only verifies a single point (4, -4) satisfies the condition, which is insufficient to determine the unique parabola.gpt-oss:20b: fail (misleading) 2026-10-16 — The sentence claims that a parabola with vertex at the origin has equations y²=4px or x²=4py, which is incorrect. For a vertical axis the correct form is (x-h)²=4p(y-k). The solution also does not address the sign of p for a parabola opening downward, leading to a misleading statement.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to derive the equation of the parabola. It states the definition and then checks a single point, which does not constitute finding the equation. The stated answer is correct, but the provided steps do not lead to it.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.