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Conics in polar form

Problem 8.543 · medium

For the conic \( \displaystyle r = \frac{8}{6 \cos{\left(\theta \right)} + 4} \), 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
\[ \frac{2}{\frac{3 \cos{\left(\theta \right)}}{2} + 1} = \frac{8}{6 \cos{\left(\theta \right)} + 4} \]
The standard form.✓ Proved
\[ \frac{\left|{6}\right|}{4} = \frac{3}{2} \]
e = 3/2: > 1, a hyperbola.✓ Proved
\[ \frac{8}{4 \cdot \frac{3}{2}} = \frac{4}{3} \]
ed = 2, so d = 4/3.✓ Proved

Answer \( e = \frac{3}{2},\ \text{hyperbola},\ \text{directrix } x = 4/3 \)

✓ 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 identifies the eccentricity and conic type. The calculation for the directrix distance d is correct, and the use of cosine implies a vertical directrix at x = d, which matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly identifies the eccentricity and conic type. The calculation for the directrix distance d is correct, and the use of cosine implies a vertical directrix at x = d, which matches the stated answer.
gpt-oss:20b: pass 2026-11-05
qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly transforms the equation into standard polar form, identifies the eccentricity and conic type, and calculates the directrix distance accurately.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly transforms the equation into standard polar form, identifies the eccentricity and conic type, and calculates the directrix distance accurately.
gpt-oss:20b: pass 2026-11-04

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.