Conics in polar form
Problem 8.316 · medium
For the conic \( \displaystyle r = \frac{8}{5 \sin{\left(\theta \right)} + 4} \), 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).
- \[ \frac{2}{\frac{5 \sin{\left(\theta \right)}}{4} + 1} = \frac{8}{5 \sin{\left(\theta \right)} + 4} \]The standard form.✓ Proved
- \[ \frac{\left|{5}\right|}{4} = \frac{5}{4} \]e = 5/4: > 1, a hyperbola.✓ Proved
- \[ \frac{8}{4 \cdot \frac{5}{4}} = \frac{8}{5} \]ed = 2, so d = 8/5.✓ Proved
Answer \( e = \frac{5}{4},\ \text{hyperbola},\ \text{directrix } y = 8/5 \)
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 (error) — The first sentence incorrectly states the standard polar form: it should be r = ed/(1 ± e sin θ) for a directrix horizontal, not cos θ. This misidentifies the form and could mislead a student.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as y = 8/5. Since the standard form involves sin(theta), the directrix is horizontal (y = d), but t
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as y = 8/5. Since the standard form involves sin(theta), the directrix is horizontal (y = d), but tgpt-oss:20b: fail (error) 2026-10-17 — The first sentence incorrectly states the standard polar form: it should be r = ed/(1 ± e sin θ) for a directrix horizontal, not cos θ. This misidentifies the form and could mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution correctly identifies the eccentricity and type, but fails to specify the orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = k), not vertical. The stated answer 'y = 8/5' is correct, but the derivation in step 4 only calculates the distance d without linking the sine term to the y-axis, which is a critical part of 'finding the directrix'.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence incorrectly states that the standard polar form is r = ed/(1 ± e cos θ). For a directrix of the form y = d the correct form is r = ed/(1 ± e sin θ). This misidentification of the standard form leads to an incorrect setup of the problem.qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is identified as y = 8/5, but the standard form involves sin(theta), which corresponds to a horizontal directrix y = d. However, the sgpt-oss:20b: fail (error) 2026-10-14 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ) when the given denominator involves sinθ, so the correct form is r = ed/(1 ± e sinθ). This misidentifies the orientation of the directrix and could mislead a student.
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.