Equations of conic sections
Problem 8.309 · easy
Find the equation of the parabola with focus \( \displaystyle (0, 4) \) and directrix \( \displaystyle y = -4 \).
- 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.
- \[ \left. \sqrt{x^{2} + \left(y - 4\right)^{2}} \right|_{\substack{ x=16\\ y=16 }} = \left. \left|{y + 4}\right| \right|_{\substack{ x=16\\ y=16 }} \]The point (16, 16) on it is equally far from focus and directrix.✓ Proved
Answer \( x^{2} = 16 y \)
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 solution only verifies that the point (16,16) satisfies the distance condition; it never derives the general equation of the parabola. The setup is correct, but the conclusion is missing.qwen3.6:27b-mlx: fail (error) — The solution fails to derive the equation of the parabola. It only verifies that a single point (16, 16) satisfies the definition, which does not constitute finding the general equation x^2 = 16y.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to derive the equation of the parabola. It only verifies that a single point (16, 16) satisfies the definition, which does not constitute finding the general equation x^2 = 16y.gpt-oss:20b: fail (error) 2026-10-17 — The solution only verifies that the point (16,16) satisfies the distance condition; it never derives the general equation of the parabola. The setup is correct, but the conclusion is missing.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to derive the equation of the parabola. It states the definition and verifies a single point, but does not perform the algebraic derivation (setting distance to focus equal to distance to directrix for a general point (x,y)) required to find the equation.gpt-oss:20b: fail (misleading) 2026-10-15 — The sentence incorrectly states the standard forms: it claims y²=4px for a vertical parabola, which is wrong. The correct form for a vertical parabola is (x-h)²=4p(y-k). The second part is correct, but the first part misleads a student about the standard equations.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 (16, 16) without deriving it, and the equation setup is presented as a check rather than a derivation.gpt-oss:20b: fail (misleading) 2026-10-14 — The solution introduces an arbitrary point (16,16) without justification and does not derive the parabola’s equation from the focus–directrix definition. This could mislead a student into thinking any point can be chosen to verify the equation, which is incorrect.
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.