Conics in polar form
Problem 8.438 · medium
For the conic \( \displaystyle r = \frac{8}{2 - 4 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{4}{1 - 2 \sin{\left(\theta \right)}} = \frac{8}{2 - 4 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-4}\right|}{2} = 2 \]e = 2: > 1, a hyperbola.✓ Proved
- \[ \frac{8}{2 \cdot 2} = 2 \]ed = 4, so d = 2.✓ Proved
Answer \( e = 2,\ \text{hyperbola},\ \text{directrix } y = -2 \)
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 (misleading) — The first sentence incorrectly states the standard form as r = ed/(1 ± e cos θ); the given equation involves sin θ, not cos θ, and the sign convention for the directrix is misrepresented. This could mislead a student about the correct setup for a polar conic with a horizontal directrix.qwen3.6:27b-mlx: fail (error) — The solution correctly identifies e=2 and d=2, but fails to determine the correct orientation of the directrix. The term -4 sin(theta) indicates the directrix is horizontal (y = constant), not vertical. Furthermore, the standard form r = ed/(1 - e sin theta) corresponds to a directrix at y = -d, so the directrix is y = -2. The stated answer claims y = -2, which is actually correct for this specific equation, but the solution text does not explicitly derive the sign/orientation logic, merely calculating d. However, looking closely at step 1, it says 'Divide... to get r = ed/(1 ± e cos θ)'. This is a generic template that ignores the sine term present in the problem. The solution then proceeds with sine in step 2, but the initial setup sentence is misleading/wrong because it specifies cosine. More critically, does the solution explicitly state the directrix is y=-2? The stated answer says y=-2. The solution calculates d=2. It does not explicitly write 'directrix is y = -2' in the sentences, but the stated answer is provided separately. The prompt asks to review the solution lines. Line 1 is factually incorrect regarding the form (claims cos, problem has sin). This is an error in the setup description.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution correctly identifies e=2 and d=2, but fails to determine the correct orientation of the directrix. The term -4 sin(theta) indicates the directrix is horizontal (y = constant), not vertical. Furthermore, the standard form r = ed/(1 - e sin theta) corresponds to a directrix at y = -d, so the directrix is y = -2. The stated answer claims y = -2, which is actually correct for this specific equation, but the solution text does not explicitly derive the sign/orientation logic, merely calculating d. However, looking closely at step 1, it says 'Divide... to get r = ed/(1 ± e cos θ)'. This is a generic template that ignores the sine term present in the problem. The solution then proceeds with sine in step 2, but the initial setup sentence is misleading/wrong because it specifies cosine. More critically, does the solution explicitly state the directrix is y=-2? The stated answer says y=-2. The solution calculates d=2. It does not explicitly write 'directrix is y = -2' in the sentences, but the stated answer is provided separately. The prompt asks to review the solution lines. Line 1 is factually incorrect regarding the form (claims cos, problem has sin). This is an error in the setup description.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence incorrectly states the standard form as r = ed/(1 ± e cos θ); the given equation involves sin θ, not cos θ, and the sign convention for the directrix is misrepresented. This could mislead a student about the correct setup for a polar conic with a horizontal directrix.qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution correctly identifies the eccentricity and the type of conic, but it fails to determine the correct directrix. The standard form derivedgpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly states the standard form with γ instead of σ, and the last line miscomputes ed (it should be 4, not 2).qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as y = -2. The standard form derived is r = 4 / (1 - 2 sin(theta)), which corresponds to a directrgpt-oss:20b: pass 2026-10-15
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.