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Conics in polar form

Problem 8.324 · medium

For the conic \( \displaystyle r = \frac{10}{6 \cos{\left(\theta \right)} + 3} \), 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).
Reviewed
\[ \frac{10}{3 \left(2 \cos{\left(\theta \right)} + 1\right)} = \frac{10}{6 \cos{\left(\theta \right)} + 3} \]
The standard form.✓ Proved
\[ \frac{\left|{6}\right|}{3} = 2 \]
e = 2: > 1, a hyperbola.✓ Proved
\[ \frac{10}{2 \cdot 3} = \frac{5}{3} \]
ed = 10/3, so d = 5/3.✓ Proved

Answer \( e = 2,\ \text{hyperbola},\ \text{directrix } x = 5/3 \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly manipulates the polar equation into standard form, identifies the eccentricity and conic type, and calculates the directrix distance accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly manipulates the polar equation into standard form, identifies the eccentricity and conic type, and calculates the directrix distance accurately.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly manipulates the equation into standard polar form, accurately identifies the eccentricity and conic type, and correctly calculates the directrix distance.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly manipulates the polar equation into standard form, identifies the eccentricity and conic type, and calculates the directrix location accurately.
gpt-oss:20b: pass 2026-10-14

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.