Equations of conic sections
Problem 8.330 · 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. A reviewer's objection to the explanation stands; see 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 first sentence incorrectly states the standard forms for a parabola. It claims that with the vertex at the origin the equation is y²=4px (or x²=4py), which mixes up the horizontal and vertical orientations. It also omits the signed distance p (here p=−3) needed to obtain the correct equation x²=−12y.qwen3.6:27b-mlx: fail (error) — The solution fails to derive the equation of the parabola. It only verifies that a single point satisfies the definition, which does not constitute finding the equation. The stated answer is correct, but the provided steps do not lead to it.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to derive the equation of the parabola. It only verifies that a single point 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 (error) 2026-10-18 — The first sentence incorrectly states the standard forms for a parabola. It claims that with the vertex at the origin the equation is y²=4px (or x²=4py), which mixes up the horizontal and vertical orientations. It also omits the signed distance p (here p=−3) needed to obtain the correct equation x²=−12y.qwen3.6:27b-mlx: fail (error) 2026-10-15 — 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 equation from the definition; it merely verifies a single point satisfies the condition, which is logically insufficient to determine the unique parabola.gpt-oss:20b: fail (error) 2026-10-15 — The solution incorrectly assumes a random point (12,-12) lies on the parabola without justification, and the algebraic substitution step is not supported by any prior result. The setup sentence also mixes up the standard forms for vertical and horizontal parabolas, which could mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to derive the equation of the parabola. Instead, it merely verifies that a single point (12, -12) satisfies the geometric 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) 2026-10-14 — Sentence 1 incorrectly states the standard form of a vertical parabola; it mixes up y²=4px and x²=4py and ignores the sign of p. The correct equation is y=x²/(4p) with p=-3, leading to x²=-12y. The rest of the solution is not affected by this mistake.
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.