Conics in polar form
Problem 8.548 · medium
For the conic \( \displaystyle r = \frac{6}{5 \sin{\left(\theta \right)} + 3} \), 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).
- \[ \frac{2}{\frac{5 \sin{\left(\theta \right)}}{3} + 1} = \frac{6}{5 \sin{\left(\theta \right)} + 3} \]The standard form.✓ Proved
- \[ \frac{\left|{5}\right|}{3} = \frac{5}{3} \]e = 5/3: > 1, a hyperbola.✓ Proved
- \[ \frac{6}{3 \cdot \frac{5}{3}} = \frac{6}{5} \]ed = 2, so d = 6/5.✓ Proved
Answer \( e = \frac{5}{3},\ \text{hyperbola},\ \text{directrix } y = 6/5 \)
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 claims the standard polar form is r = ed/(1 ± e cos θ); the given conic involves sin θ, so the form should be r = ed/(1 ± e sin θ). This misstates the model and could mislead a student.qwen3.6:27b-mlx: fail (error) — The solution fails to identify the directrix as a horizontal line (y = d) because the denominator contains sin(theta), not cos(theta). It also incorrectly states ed = 2 in the final step, whereas ed = 6/3 = 2 is correct, but the calculation of d is derived from ed=2 and e=5/3, yielding d=6/5, which is numerically correct but the text 'ed=2' is confusingly presented alongside the fraction arithmetic. The primary error is the missing specification of the directrix's orientation (y vs x).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-05 — The solution fails to identify the directrix as a horizontal line (y = d) because the denominator contains sin(theta), not cos(theta). It also incorrectly states ed = 2 in the final step, whereas ed = 6/3 = 2 is correct, but the calculation of d is derived from ed=2 and e=5/3, yielding d=6/5, which is numerically correct but the text 'ed=2' is confusingly presented alongside the fraction arithmetic. The primary error is the missing specification of the directrix's orientation (y vs x).gpt-oss:20b: fail (error) 2026-11-05 — The first sentence incorrectly claims the standard polar form is r = ed/(1 ± e cos θ); the given conic involves sin θ, so the form should be r = ed/(1 ± e sin θ). This misstates the model and could mislead a student.qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution fails to identify the directrix as a horizontal line (y = d) because the denominator contains sin(theta). It incorrectly implies the standard form is r = ed/(1 ± e cos θ) which corresponds to vertical directrices, leading to an incomplete or potentially confusing identification of the directrix location.gpt-oss:20b: fail (error) 2026-11-04 — Sentence 1 incorrectly states the standard form as r = ed/(1 ± e cosθ) when the given polar equation involves sinθ, and it does not correctly describe the division process or the sign convention. The subsequent algebraic steps are fine, but the initial sentence misleads the reader about the correct form of the polar conic.qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution fails to identify the correct orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = d), but the standard form derivation implies the directrix is y = 6/5. However, the sign in the denominator is positive (+ 3), which corresponds to the form r = ed / (1 + e sin theta). For this form, the directrix is y = +d. The calculation d = 6/5 is correct for the distance, but the sentence in step 1 claims the target form is r = ed/(1 ± e cos theta), which is incorrect for a sine term. This mismatch between the stated method (cosine) and the problem (sine) is a significant error in reasoning, even if the final numerical answer for the directrix location happens to be correct by coincidence of the sign handling in step 4. More critically, step 1 explicitly states the wrong standard form template.gpt-oss:20b: fail (misleading) 2026-11-04 — The first sentence incorrectly states the standard form as involving cos θ, whereas the given polar equation uses sin θ. This could mislead a student into applying the wrong standard form.
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.