Conics in polar form
Problem 8.552 · medium
For the conic \( \displaystyle r = \frac{2}{5 - 8 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{2}{5 \left(1 - \frac{8 \sin{\left(\theta \right)}}{5}\right)} = \frac{2}{5 - 8 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-8}\right|}{5} = \frac{8}{5} \]e = 8/5: > 1, a hyperbola.✓ Proved
- \[ \frac{2}{5 \cdot \frac{8}{5}} = \frac{1}{4} \]ed = 2/5, so d = 1/4.✓ Proved
Answer \( e = \frac{8}{5},\ \text{hyperbola},\ \text{directrix } y = -1/4 \)
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) — Sentence 1 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).qwen3.6:27b-mlx: fail (error) — The solution fails to account for the sign of the sine term in the denominator. The form r = ed / (1 - e sin(theta)) corresponds to a directrix at y = -d, not y = d. Since the standard form derived has a negative sine term (after rearranging to 1 - (8/5)sin(theta)), the directrix is y = -1/4, but the solution's logic for determining the directrix location from the sign is missing or implicit, leading to a potential confusion if the student assumes positive d always means positive y. More critically, step 1 says 'divide... to get r = ed/(1 ± e cos θ)' but the problem involves sin, not cos. This is a significant error in the stated method.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-05 — The solution fails to account for the sign of the sine term in the denominator. The form r = ed / (1 - e sin(theta)) corresponds to a directrix at y = -d, not y = d. Since the standard form derived has a negative sine term (after rearranging to 1 - (8/5)sin(theta)), the directrix is y = -1/4, but the solution's logic for determining the directrix location from the sign is missing or implicit, leading to a potential confusion if the student assumes positive d always means positive y. More critically, step 1 says 'divide... to get r = ed/(1 ± e cos θ)' but the problem involves sin, not cos. This is a significant error in the stated method.gpt-oss:20b: fail (misleading) 2026-11-05 — Sentence 1 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).qwen3.6:27b-mlx: inconclusive 2026-11-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is identified as y = -1/4, but the negative sign in the denominator (5 - 8 sin(theta)) implies the directrix is below the pole, so itgpt-oss:20b: fail (error) 2026-11-04 — The first sentence incorrectly states the standard form as involving θ with α cosine term and a ± sign, whereas the given polar equation uses a sine term and a minus sign. It also misidentifies the constant to divide by and does not correctly derive the parameters e and d from the given equation.qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution fails to account for the negative sign in the denominator, leading to an incorrect directrix. For r = ed / (1 - e sin theta), the directrix is y = -d. Since d = 1/4, the directrix should be y = -1/4, but the derivation in step 4 implies d=1/4 without specifying the sign convention relative to the sine term, potentially confusing the student about why the directrix is negative. More critically, the standard form conversion in step 2 results in 1 - (8/5)sin(theta), which corresponds to a directrix at y = -d. The stated answer is correct, but the reasoning in step 4 is incomplete as it calculates d but does not explicitly link the minus sign in the denominator to the negative y-coordinate of the directrix.gpt-oss:20b: fail (error) 2026-11-04 — The first sentence incorrectly states that dividing by the constant in the denominator yields the standard form r = ed/(1 ± e cosθ). The given polar equation involves sinθ, not cosθ, so the standard form should be r = ed/(1 ± e sinθ). This misidentification of the standard form leads to an incorrect setup and consequently to an incorrect determination 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 2026-11-05 with SymPy 1.14.0.