Conics in polar form
Problem 8.461 · medium
For the conic \( \displaystyle r = \frac{11}{8 \sin{\left(\theta \right)} + 3} \), 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).
- \[ \frac{11}{3 \left(\frac{8 \sin{\left(\theta \right)}}{3} + 1\right)} = \frac{11}{8 \sin{\left(\theta \right)} + 3} \]The standard form.✓ Proved
- \[ \frac{\left|{8}\right|}{3} = \frac{8}{3} \]e = 8/3: > 1, a hyperbola.✓ Proved
- \[ \frac{11}{3 \cdot \frac{8}{3}} = \frac{11}{8} \]ed = 11/3, so d = 11/8.✓ Proved
Answer \( e = \frac{8}{3},\ \text{hyperbola},\ \text{directrix } y = 11/8 \)
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 first sentence incorrectly states the standard form uses cos θ, whereas the given equation involves sin θ. This could mislead a student about the correct polar form for a conic with a vertical directrix.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as y = 11/8. Since the standard form involves sin(theta), the directrix is horizontal (y = d), but
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as y = 11/8. Since the standard form involves sin(theta), the directrix is horizontal (y = d), butgpt-oss:20b: fail (misleading) 2026-10-19 — The first sentence incorrectly states the standard form uses cos θ, whereas the given equation involves sin θ. This could mislead a student about the correct polar form for a conic with a vertical directrix.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution correctly identifies the eccentricity and the type of conic, but fails to specify the orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = d), not vertical. The stated answer 'y = 11/8' is correct, but the derivation in step 4 does not justify why the directrix is y = d rather than x = d or y = -d, relying on an implicit assumption not stated in the sentences.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as y = 11/8. Since the denominator contains sin(theta), the directrix is horizontal, but the standgpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence incorrectly states the standard form as r = ed/(1 ± e cosθ); the given equation involves sinθ, so the correct form is r = ed/(1 ± e sinθ).
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.