Conics in polar form
Problem 8.459 · medium
For the conic \( \displaystyle r = \frac{9}{1 - 5 \cos{\left(\theta \right)}} \), 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).
- \[ \frac{9}{1 \left(- \frac{5 \cos{\left(\theta \right)}}{1} + 1\right)} = \frac{9}{1 - 5 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-5}\right|}{1} = 5 \]e = 5: > 1, a hyperbola.✓ Proved
- \[ \frac{9}{1 \cdot 5} = \frac{9}{5} \]ed = 9, so d = 9/5.✓ Proved
Answer \( e = 5,\ \text{hyperbola},\ \text{directrix } x = -9/5 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 solution claims the directrix is x = -9/5, but for the polar equation r = ed/(1 - e cosθ) the directrix is x = +d, so the correct directrix is x = 9/5. The sign error would mislead a student.qwen3.6:27b-mlx: fail (error) — The solution fails to determine the correct sign of the directrix. For the form r = ed / (1 - e cos theta), the directrix is x = -d, so it should be x = -9/5. The stated answer is correct, but the derivation in step 4 only calculates the magnitude d and does not justify the negative sign required by the minus in the denominator.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to determine the correct sign of the directrix. For the form r = ed / (1 - e cos theta), the directrix is x = -d, so it should be x = -9/5. The stated answer is correct, but the derivation in step 4 only calculates the magnitude d and does not justify the negative sign required by the minus in the denominator.gpt-oss:20b: fail (misleading) 2026-10-19 — The solution claims the directrix is x = -9/5, but for the polar equation r = ed/(1 - e cosθ) the directrix is x = +d, so the correct directrix is x = 9/5. The sign error would mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution correctly identifies e=5 and d=9/5, but fails to determine the correct sign for the directrix. The form r = ed/(1 - e cos theta) corresponds to a directrix at x = -d, so the directrix is x = -9/5. While the stated answer is correct, the solution steps do not derive this sign, leaving a gap in the logic regarding how the minus sign in the denominator maps to the directrix location.gpt-oss:20b: fail (error) 2026-10-16 — The directrix is incorrectly given as x = -9/5. For r = ed/(1 - e cosθ) with e > 1, the directrix is x = +d = 9/5, not -9/5.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution correctly identifies e=5 and d=9/5, but fails to determine the correct sign for the directrix. The form r = ed/(1 - e cos theta) corresponds to a directrix at x = -d, so the directrix is x = -9/5. While the stated answer is correct, the solution steps do not justify the negative sign, leaving the student with the ambiguous value d=9/5 without linking it to the specific geometry (left vs right directrix) implied by the minus sign in the denominator.gpt-oss:20b: fail (misleading) 2026-10-15 — The solution correctly identifies e=5 and that the conic is a hyperbola, but it incorrectly states the directrix as x = -9/5. For the polar equation r = ed/(1 – e cosθ), the directrix is x = +d = 9/5, not –9/5. This would mislead a student about the location of the directrix.
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.