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Conics in polar form

Problem 8.437 · medium

For the conic \( \displaystyle r = \frac{8}{5 - 6 \cos{\left(\theta \right)}} \), 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).
\[ \frac{8}{5 \left(1 - \frac{6 \cos{\left(\theta \right)}}{5}\right)} = \frac{8}{5 - 6 \cos{\left(\theta \right)}} \]
The standard form.✓ Proved
\[ \frac{\left|{-6}\right|}{5} = \frac{6}{5} \]
e = 6/5: > 1, a hyperbola.✓ Proved
\[ \frac{8}{5 \cdot \frac{6}{5}} = \frac{4}{3} \]
ed = 8/5, so d = 4/3.✓ Proved

Answer \( e = \frac{6}{5},\ \text{hyperbola},\ \text{directrix } x = -4/3 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to determine the correct sign of the directrix. The standard form r = ed / (1 - e cos(theta)) corresponds to a directrix at x = -d. Since the denominator is 1 - (6/5)cos(theta), the directrix is x = -4/3, but the solution does not explicitly link the minus sign in the denominator to the negative x-coordinate, risking confusion with the form 1 + e cos(theta) which would yield x = d.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to determine the correct sign of the directrix. The standard form r = ed / (1 - e cos(theta)) corresponds to a directrix at x = -d. Since the denominator is 1 - (6/5)cos(theta), the directrix is x = -4/3, but the solution does not explicitly link the minus sign in the denominator to the negative x-coordinate, risking confusion with the form 1 + e cos(theta) which would yield x = d.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to determine the correct sign of the directrix. The standard form with a negative cosine term, r = ed/(1 - e cos θ), corresponds to a directrix at x = -d. Since d = 4/3, the directrix is x = -4/3. While the final answer happens to be correct, the derivation in step 4 only calculates the magnitude d and does not justify the negative sign required by the form 1 - e cos θ.
gpt-oss:20b: fail (misleading) 2026-10-16 — The solution correctly finds e=6/5 and d=4/3, but it incorrectly assigns the directrix as x=-4/3. For the form r=ed/(1‑e cosθ) the directrix is x=+d, so the correct directrix is x=4/3, not -4/3.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to account for the negative sign in the denominator term (-6 cos(theta)). The standard form r = ed/(1 - e cos(theta)) corresponds to a directrix at x = -d. Since d = 4/3, the directrix is x = -4/3. While the final numerical answer for the directrix is correct, the reasoning in step 4 implicitly assumes the form 1 + e cos(theta) (which would yield x = +d) or ignores the sign convention entirely, leading to a logically flawed derivation of the directrix location relative to the standard forms.
gpt-oss:20b: fail (error) 2026-10-15 — The directrix sign is incorrect. For r = ed/(1‑e cosθ) the directrix is x = +d, not x = –d. The solution should state x = 4/3.

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 2026-10-18 with SymPy 1.14.0.