Conics in polar form
Problem 8.547 · medium
For the conic \( \displaystyle r = \frac{6}{2 - 4 \cos{\left(\theta \right)}} \), 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).
- \[ \frac{3}{1 - 2 \cos{\left(\theta \right)}} = \frac{6}{2 - 4 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-4}\right|}{2} = 2 \]e = 2: > 1, a hyperbola.✓ Proved
- \[ \frac{6}{2 \cdot 2} = \frac{3}{2} \]ed = 3, so d = 3/2.✓ Proved
Answer \( e = 2,\ \text{hyperbola},\ \text{directrix } x = -3/2 \)
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 directrix should be x = +3/2, not x = -3/2. The solution’s conclusion about the directrix is incorrect.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as x = -3/2. The standard form derived is r = 3 / (1 - 2 cos(theta)), which corresponds to a direc
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 = -3/2. The standard form derived is r = 3 / (1 - 2 cos(theta)), which corresponds to a direcgpt-oss:20b: fail (misleading) 2026-11-05 — The directrix should be x = +3/2, not x = -3/2. The solution’s conclusion about the directrix is incorrect.qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution fails to account for the sign change in the denominator when converting to standard form. Dividing by 2 yields r = 3 / (1 - 2 cos(theta)), which corresponds to a directrix at x = -d = -3/2. However, the calculation in step 4 treats 'ed' as positive 3/2 without explicitly linking the negative sign in the denominator to the negative x-coordinate of the directrix, potentially confusing the student about the relationship between the sign in the polar equation and the directrix location.gpt-oss:20b: fail (error) 2026-11-04 — The directrix sign is incorrect; for r = ed/(1‑e cosθ) the directrix is x = +d, not –d. The solution’s conclusion about the directrix is false.qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution fails to account for the negative sign in the denominator. The standard form r = ed / (1 - e cos θ) corresponds to a directrix at x = -d. Since e=2 and d=3/2, the directrix is x = -3/2, but the derivation in step 4 ignores the sign change from dividing by 2, leading to a potential confusion about whether the directrix is x = 3/2 or x = -3/2 without explicit justification of the form used.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.