Conics in polar form
Problem 8.315 · medium
For the conic \( \displaystyle r = \frac{5}{1 - 6 \cos{\left(\theta \right)}} \), 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).
- \[ \frac{5}{1 \left(- \frac{6 \cos{\left(\theta \right)}}{1} + 1\right)} = \frac{5}{1 - 6 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-6}\right|}{1} = 6 \]e = 6: > 1, a hyperbola.✓ Proved
- \[ \frac{5}{1 \cdot 6} = \frac{5}{6} \]ed = 5, so d = 5/6.✓ Proved
Answer \( e = 6,\ \text{hyperbola},\ \text{directrix } x = -5/6 \)
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: fail (error) — The directrix is incorrectly given. For r = ed/(1 – e cosθ) the directrix lies at x = +d, so it should be x = 5/6, not x = –5/6.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The directrix is identified as x = -5/6, but the standard form r = ed/(1 - e cos theta) corresponds to a vertical directrix at x = d (positive). Since d = 5/6, the directrix should be x = 5/6. The negative sign in the denominator indicates the directrix is to the right of the pole, not left.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The directrix is identified as x = -5/6, but the standard form r = ed/(1 - e cos theta) corresponds to a vertical directrix at x = d (positive). Since d = 5/6, the directrix should be x = 5/6. The negative sign in the denominator indicates the directrix is to the right of the pole, not left.gpt-oss:20b: fail (error) 2026-10-17 — The directrix is incorrectly given. For r = ed/(1 – e cosθ) the directrix lies at x = +d, so it should be x = 5/6, not x = –5/6.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The directrix is identified as x = -5/6, but for the form r = ed/(1 - e cos θ), the directrix is x = -d, which is x = -5/6. Wait, let me re-verify. Standard form r = ed / (1 - e cos theta) corresponds to directrix x = -d. Here e=6, ed=5 => d=5/6. So directrix is x = -5/6. The stated answer says x = -5/6. The solution steps calculate d=5/6 but do not explicitly state the sign rule for the directrix location relative to the denominator sign. However, the final answer in the problem statement is x = -5/6. Let's look closer at the standard forms. r = ep / (1 - e cos theta) -> directrix x = -p. r = ep / (1 + e cos theta) -> directrix x = p. Here we have 1 - 6 cos theta, so directrix is x = -d = -5/6. The solution calculates d=5/6. It does not explicitly write the final coordinate, but the 'Stated answer' is separate. The solution lines themselves end at calculating d. Does the solution conclude the directrix? Line 4 says 'ed = 5, so d = 5/6'. It does not state the directrix equation. The problem asks to 'find the directrix'. The solution fails to state the final directrix equation, only finding the distance d. Furthermore, usually 'd' or 'p' is the distance from the focus to the directrix. The directrix is the line x = -5/6. The solution stops short of stating the directrix.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The directrix is incorrectly identified as x = -5/6. For the form r = ed / (1 - e cos(theta)), the directrix is x = -d, which is x = -5/6. Wait, let me re-verify. Standard form r = ed / (1 - e cos theta) corresponds to directrix x = -d. Here e=6, ed=5 => d=5/6. So directrix is x = -5/6. The stated answer says x = -5/6. The solution steps calculate d=5/6 but do not explicitly state the sign or the equation of the directrix, just 'd=5/6'. However, the prompt asks to review the *solution* lines. Line 4 calculates d. It does not state the directrix equation. The 'Stated answer' is separate. The solution lines themselves are algebraically correct derivations of e and d. But does it identify the directrix? The problem asks to 'find the directrix'. The solution stops at finding d. It fails to state the final answer for the directrix. Also, line 1 says 'Divide top and bottom...'. The given equation is already in the form 5/(1-6cos). No division is needed. This is a misleading instruction. Furthermore, usually standard form is r = ep / (1 +/- e cos theta). The sign in the denominator determines the directrix. 1 - 6 cos theta implies directrix x = -p. The solution finds p=5/6 but doesn't conclude x = -5/6. It is incomplete.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.