Equations of conic sections
Problem 8.300 · easy
Find the equation of the parabola with focus \( \displaystyle (0, 3) \) and directrix \( \displaystyle y = -3 \).
- 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.
- \[ \left. \sqrt{x^{2} + \left(y - 3\right)^{2}} \right|_{\substack{ x=12\\ y=12 }} = \left. \left|{y + 3}\right| \right|_{\substack{ x=12\\ y=12 }} \]The point (12, 12) on it is equally far from focus and directrix.✓ Proved
Answer \( x^{2} = 12 y \)
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 (misleading) — The solution never derives the equation from the given data; it merely states that a single point satisfies the distance condition, which does not establish the parabola’s equation. The reasoning is incomplete and could mislead a student into thinking that checking one point is sufficient.qwen3.6:27b-mlx: pass — The solution correctly identifies the orientation and parameters of the parabola, leading to the correct equation x^2 = 12y. The verification step is valid.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the orientation and parameters of the parabola, leading to the correct equation x^2 = 12y. The verification step is valid.gpt-oss:20b: fail (misleading) 2026-10-17 — The solution never derives the equation from the given data; it merely states that a single point satisfies the distance condition, which does not establish the parabola’s equation. The reasoning is incomplete and could mislead a student into thinking that checking one point is sufficient.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to derive the equation of the parabola. It only verifies that a single point (12, 12) satisfies the definition, which does not constitute finding the equation. The stated answer is correct, but the provided steps do not lead to it.gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence incorrectly states the standard forms for a parabola. It claims both y²=4px and x²=4py apply, but only x²=4py is relevant for a vertical parabola with vertex at the origin. It also omits the key step of computing p=3 (the distance from the vertex to the focus) and does not justify the chosen orientation. This could mislead a student into using the wrong form or forgetting to determine p.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the standard form as y² = 4px for a vertical parabola; it should be x² = 4py. Additionally, the verification step uses an arbitrary point (12, 12) which is not on the parabola (12² ≠ 12*12 is false, wait 144=144, but the derivation logic is flawed because it doesn't derive the equation, just checks a point that happens to work for the final answer but the setup text is wrong).gpt-oss:20b: fail (misleading) 2026-10-14 — The solution’s second sentence merely plugs an arbitrary point (12,12) into the distance equality without deriving the parabola’s equation or explaining why that point is chosen. It does not show how the standard form x²=4py follows from the focus and directrix, leaving the conclusion unsupported and potentially confusing to a student.
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 2026-10-17 with SymPy 1.14.0.