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Conics in polar form

Problem 8.312 · medium

For the conic \( \displaystyle r = \frac{3}{5 - 5 \sin{\left(\theta \right)}} \), 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).
\[ \frac{3}{5 \left(1 - \sin{\left(\theta \right)}\right)} = \frac{3}{5 - 5 \sin{\left(\theta \right)}} \]
The standard form.✓ Proved
\[ \frac{\left|{-5}\right|}{5} = 1 \]
e = 1: = 1, a parabola.✓ Proved
\[ \frac{3}{1 \cdot 5} = \frac{3}{5} \]
ed = 3/5, so d = 3/5.✓ Proved

Answer \( e = 1,\ \text{parabola},\ \text{directrix } y = -3/5 \)

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 first sentence incorrectly states the standard form as r = ed/(1 ± e cosθ) when the given equation involves sinθ. This could mislead a student into using the wrong form for the directrix and orientation.
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution correctly identifies e=1 and d=3/5, but fails to determine the correct orientation of the directrix. Since the term is -sin(theta), the

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 correctly identifies e=1 and d=3/5, but fails to determine the correct orientation of the directrix. Since the term is -sin(theta), the
gpt-oss:20b: fail (misleading) 2026-10-17 — The first sentence incorrectly states the standard form as r = ed/(1 ± e cosθ) when the given equation involves sinθ. This could mislead a student into using the wrong form for the directrix and orientation.
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is incorrectly identified as y = -3/5. For the form r = ed / (1 - e sin(theta)), the directrix is y = -d. Since d = 3/5, the directrix
gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ); for the given equation the denominator involves sinθ, not cosθ. This misleads a student into using the wrong form and could cause confusion about the orientation of the directrix.
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution correctly identifies e=1 and d=3/5, but fails to determine the correct sign and orientation for the directrix. The term -5 sin(theta) implies a directrix of the form y = -d, so the directrix is y = -3/5. However, the solution text does not explicitly derive the sign from the negative sine term, merely stating 'd = 3/5' and then jumping to the final answer. While the final answer happens to be correct, the reasoning presented in step 4 is incomplete and potentially misleading regarding how the sign of the directrix is determined from the standard form r = ed / (1 - e sin theta).
gpt-oss:20b: fail (misleading) 2026-10-14 — The first sentence incorrectly claims the standard form involves cos θ, whereas the given equation involves sin θ. This misstates the orientation and could mislead a student about the correct polar form of the conic.

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.