Conics in polar form
Problem 8.454 · medium
For the conic \( \displaystyle r = \frac{6}{6 \cos{\left(\theta \right)} + 1} \), 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).Reviewed
- \[ \frac{6}{1 \left(\frac{6 \cos{\left(\theta \right)}}{1} + 1\right)} = \frac{6}{6 \cos{\left(\theta \right)} + 1} \]The standard form.✓ Proved
- \[ \frac{\left|{6}\right|}{1} = 6 \]e = 6: > 1, a hyperbola.✓ Proved
- \[ \frac{6}{1 \cdot 6} = 1 \]ed = 6, so d = 1.✓ Proved
Answer \( e = 6,\ \text{hyperbola},\ \text{directrix } x = 1 \)
Lines: 3 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the eccentricity and conic type. It also correctly determines the directrix distance d=1 and, given the positive cosine term, implies the directrix is x=1.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the eccentricity and conic type. It also correctly determines the directrix distance d=1 and, given the positive cosine term, implies the directrix is x=1.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the standard form, calculates the eccentricity and directrix distance, and properly identifies the conic type.gpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly claims that dividing numerator and denominator by the constant in the denominator yields the standard polar form r=ed/(1±e cosθ). In fact, r=6/(6 cosθ+1)=6/(1+6 cosθ) is already in the form r=ed/(1+e cosθ) with e=6 and d=1; the division step is unnecessary and misleading. The subsequent steps then incorrectly justify e and d based on a misapplied formula.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the standard form parameters and derives the eccentricity, conic type, and directrix equation.gpt-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 2026-10-19 with SymPy 1.14.0.