Conics in polar form
Problem 8.443 · medium
For the conic \( \displaystyle r = \frac{12}{1 - \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.
- Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ).
- \[ \frac{12}{1 \left(- \frac{\sin{\left(\theta \right)}}{1} + 1\right)} = \frac{12}{1 - \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-1}\right|}{1} = 1 \]e = 1: = 1, a parabola.✓ Proved
- \[ \frac{12}{1 \cdot 1} = 12 \]ed = 12, so d = 12.✓ Proved
Answer \( e = 1,\ \text{parabola},\ \text{directrix } y = -12 \)
Lines: 3 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ 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 | for points on the curve, distance to the focus over distance to the directrix is e |
Reviewers
gpt-oss:20b: fail (error) — The first sentence incorrectly claims that dividing the numerator and denominator by the constant in the denominator yields the standard polar form r=ed/(1±e cosθ). The given conic uses sinθ, not cosθ, and the manipulation is not justified. The subsequent steps incorrectly identify e and d based on this flawed statement, leading to an incorrect conclusion about the conic type and directrix.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution correctly identifies e=1 and d=12, but fails to determine the correct orientation of the directrix. For r = ed/(1 - sin(theta)), the di
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution correctly identifies e=1 and d=12, but fails to determine the correct orientation of the directrix. For r = ed/(1 - sin(theta)), the digpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly claims that dividing the numerator and denominator by the constant in the denominator yields the standard polar form r=ed/(1±e cosθ). The given conic uses sinθ, not cosθ, and the manipulation is not justified. The subsequent steps incorrectly identify e and d based on this flawed statement, leading to an incorrect conclusion about the conic type and directrix.qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution correctly identifies the eccentricity and distance d, but fails to determine the correct position of the directrix. For the form r = ed / (1 - sin(theta)), the directrix is y = -d, so y = -12. However, the solution does not explicitly state this geometric relationship or justify why the directrix is horizontal and below the pole, merely calculating d. While the final answer matches, the reasoning is incomplete and potentially misleading regarding how the sign in the denominator maps to the directrix location.gpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly claims the standard form is r = ed/(1 ± e cosθ). The given polar equation involves sinθ, so the correct standard form is r = ed/(1 ± e sinθ). This misstates the model and could mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution correctly identifies the eccentricity and the distance d, but fails to determine the correct orientation of the directrix. For the form r = ed / (1 - sin(theta)), the directrix is y = -d, so y = -12. However, the solution steps do not explicitly state the rule linking the sign of the sine term to the position of the directrix (y = -d vs y = d), making the derivation of the directrix's location incomplete and potentially confusing for a student who might assume y = 12. More critically, standard polar forms are usually r = ed / (1 + e sin theta) for y = d and r = ed / (1 - e sin theta) for y = -d. The solution calculates d=12 but does not justify why the directrix is y = -12 rather than y = 12 based on the minus sign in the denominator.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence incorrectly states the standard polar form of a conic. It claims that dividing by the constant in the denominator yields r = ed/(1 ± e cos θ), but the given equation involves sin θ, not cos θ, and the sign convention is wrong. Consequently the subsequent identification of e, the conic type, and the directrix are based on a faulty setup.
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/polar_conics, checked 2026-10-18 with SymPy 1.14.0.