Conics in polar form
Problem 8.544 · medium
For the conic \( \displaystyle r = \frac{1}{6 \cos{\left(\theta \right)} + 2} \), 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).
- \[ \frac{1}{2 \left(3 \cos{\left(\theta \right)} + 1\right)} = \frac{1}{6 \cos{\left(\theta \right)} + 2} \]The standard form.✓ Proved
- \[ \frac{\left|{6}\right|}{2} = 3 \]e = 3: > 1, a hyperbola.✓ Proved
- \[ 1 \cdot \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \]ed = 1/2, so d = 1/6.✓ Proved
Answer \( e = 3,\ \text{hyperbola},\ \text{directrix } x = 1/6 \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as x = 1/6. Since the standard form derived is r = (ed)/(1 + e cos(theta)), the directrix is verti
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-11-05 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as x = 1/6. Since the standard form derived is r = (ed)/(1 + e cos(theta)), the directrix is vertigpt-oss:20b: pass 2026-11-05qwen3.6:27b-mlx: inconclusive 2026-11-04 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The directrix is incorrect. For the form r = ed/(1 + e cos θ), the directrix is x = d. Here d = 1/6, so the directrix is x = 1/6. Wait, let me re-evagpt-oss:20b: pass 2026-11-04qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly transforms the equation into standard polar form, accurately identifies the eccentricity and conic type, and correctly calculates the directrix.gpt-oss:20b: pass 2026-11-04
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 2026-11-05 with SymPy 1.14.0.