Equations of conic sections
Problem 8.481 · 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. 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 first sentence incorrectly states the standard forms for a parabola; it mixes the horizontal form y²=4px with the vertical form x²=4py without clarifying that the given parabola is vertical. It also omits the sign of p, which is essential for determining the direction of opening. This could mislead a student into applying the wrong equation.qwen3.6:27b-mlx: pass — The solution correctly identifies the geometric definition and the standard form for a vertical parabola with vertex at the origin. The verification step confirms the derived equation satisfies the condition for a specific point.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the geometric definition and the standard form for a vertical parabola with vertex at the origin. The verification step confirms the derived equation satisfies the condition for a specific point.gpt-oss:20b: fail (misleading) 2026-10-19 — The first sentence incorrectly states the standard forms for a parabola; it mixes the horizontal form y²=4px with the vertical form x²=4py without clarifying that the given parabola is vertical. It also omits the sign of p, which is essential for determining the direction of opening. This could mislead a student into applying the wrong equation.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly identifies the vertex as the origin. The vertex is the midpoint between the focus (0, -4) and the directrix y=4, which is (0, 0), so the vertex is indeed at the origin. However, the standard form used in step 1 is ambiguous or misapplied. More critically, step 2 checks a single point (16, -16) to verify the equation, which is not a derivation but a check. The primary error is in the reasoning: the solution claims the vertex is at the origin (which is true) but then uses a generic form without explicitly deriving p. Let's look closer. Focus (0, -4), Directrix y=4. Vertex is (0,0). Distance p from vertex to focus is 4. Since focus is below vertex, parabola opens down. Equation is x^2 = -4py => x^2 = -16y. The stated answer is correct. Step 1 says 'with vertex at the origin that gives y^2 = 4px (or x^2 = 4py)'. This is a valid setup. Step 2 is just a verification check, not a derivation step, but it is marked as an equation proved by CAS. The prompt asks to review the solution. Is there an error? The solution doesn't actually derive the answer, it just states a property and checks a point. This is not a valid derivation. A student would not learn how to find the equation. It's a 'pass' on the final answer but 'fail' on the method/logic presented as a solution. The sentence in step 1 is correct. The equation in step 2 is correct. But does it constitute a solution? No. It's incomplete. However, usually these reviews look for factual errors. Is there a factual error? No. Is it misleading? Yes, it implies checking one point is sufficient or that the setup alone yields the answer without calculating p. Actually, step 1 defines p as 'focus's distance from the vertex'. It does not calculate p. It just gives the form. Then it checks a point. It never calculates p=4. This is a severe logical gap. A student would not know how to get 16. So it is misleading/error in pedagogy.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution states the general form y²=4px (or x²=4py) but never determines the value of p from the given focus and directrix. A student could incorrectly assume any p works, so the conclusion that x²=-16y is not justified in the text.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the geometric definition and the resulting standard equation form. The unchecked equation line serves as a valid verification step for a specific point on the curve, confirming the derived equation is correct.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution incorrectly states the standard forms for a parabola with vertex at the origin. It claims "y² = 4px (or x² = 4py)" as the general equation, which is not correct for a vertical parabola opening downwards. The correct form should be y = -x²/(4p) (or equivalently x² = -4py) with p = 4. The sentence misleads a student about the relationship between the vertex, focus, and directrix.
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 2026-10-19 with SymPy 1.14.0.