Conics in polar form
Problem 8.434 · medium
For the conic \( \displaystyle r = \frac{6}{1 - 2 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{6}{1 \left(- \frac{2 \sin{\left(\theta \right)}}{1} + 1\right)} = \frac{6}{1 - 2 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-2}\right|}{1} = 2 \]e = 2: > 1, a hyperbola.✓ Proved
- \[ \frac{6}{1 \cdot 2} = 3 \]ed = 6, so d = 3.✓ Proved
Answer \( e = 2,\ \text{hyperbola},\ \text{directrix } y = -3 \)
Lines: 3 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 |
| 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 (misleading) — The first sentence incorrectly claims the standard form is r=ed/(1±e cosθ) after dividing by the constant, ignoring that the given equation involves sinθ. This misleads a student into applying the wrong standard form. The rest of the steps (e=2, hyperbola, d=3, directrix y=-3) are correct, but the erroneous sentence invalidates the solution.qwen3.6:27b-mlx: fail (error) — The solution correctly identifies e=2 and d=3, but fails to determine the correct sign for the directrix. The form r = ed/(1 - e sin theta) corresponds to a directrix at y = -d, so the directrix is y = -3. While the stated answer is correct, the solution steps do not justify the negative sign, merely calculating d=3. A student following this logic might incorrectly conclude y=3.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution correctly identifies e=2 and d=3, but fails to determine the correct sign for the directrix. The form r = ed/(1 - e sin theta) corresponds to a directrix at y = -d, so the directrix is y = -3. While the stated answer is correct, the solution steps do not justify the negative sign, merely calculating d=3. A student following this logic might incorrectly conclude y=3.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence incorrectly claims the standard form is r=ed/(1±e cosθ) after dividing by the constant, ignoring that the given equation involves sinθ. This misleads a student into applying the wrong standard form. The rest of the steps (e=2, hyperbola, d=3, directrix y=-3) are correct, but the erroneous sentence invalidates the solution.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution correctly identifies e=2 and d=3, but fails to determine the correct sign and orientation for the directrix. For r = ed/(1 - e sin theta), the directrix is y = -d, so y = -3. However, the solution does not explicitly state this derivation, merely calculating d=3. While the final answer in the 'Stated answer' is correct, the solution steps provided (1-4) do not logically lead to the specific directrix equation y = -3 without an additional step explaining the relationship between the form (1 - e sin theta) and the directrix location (y = -d). A student might incorrectly assume y = 3 or x = 3. The solution is incomplete/misleading regarding the directrix derivation.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution incorrectly states the standard form uses cosθ and misidentifies the directrix sign. The correct directrix for r=6/(1-2sinθ) is y=3, not y=-3. The setup and algebra are otherwise fine.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution correctly identifies e=2 and d=3, but fails to determine the correct sign for the directrix. The form r = ed/(1 - e sin theta) corresponds to a directrix at y = -d, so the directrix is y = -3. However, the solution text does not explicitly state the rule linking the minus sign in the denominator to the negative y-coordinate, leaving the derivation of the directrix's position incomplete and potentially confusing regarding the sign convention.gpt-oss:20b: fail (error) 2026-10-15 — Sentence 1 incorrectly states the standard form as r = ed/(1 ± e cosθ) and claims to divide by the constant in the denominator; the given equation involves sinθ, not cosθ, and no division is performed. The setup misidentifies the standard form, which could mislead 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/polar_conics, checked 2026-10-18 with SymPy 1.14.0.