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Conics in polar form

Problem 8.323 · medium

For the conic \( \displaystyle r = \frac{8}{7 \cos{\left(\theta \right)} + 1} \), 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).
\[ \frac{8}{1 \left(\frac{7 \cos{\left(\theta \right)}}{1} + 1\right)} = \frac{8}{7 \cos{\left(\theta \right)} + 1} \]
The standard form.✓ Proved
\[ \frac{\left|{7}\right|}{1} = 7 \]
e = 7: > 1, a hyperbola.✓ Proved
\[ \frac{8}{1 \cdot 7} = \frac{8}{7} \]
ed = 8, so d = 8/7.✓ Proved

Answer \( e = 7,\ \text{hyperbola},\ \text{directrix } x = 8/7 \)

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: fail (misleading) — The first sentence incorrectly describes the manipulation needed to reach the standard polar form; it suggests dividing by a constant in the denominator, which is not the correct step. The rest of the solution is fine, but the misleading statement could confuse a student about how to rewrite the given equation.
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to correctly transform the equation into standard polar form. The standard form requires the constant term in the denominator to

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 fails to correctly transform the equation into standard polar form. The standard form requires the constant term in the denominator to
gpt-oss:20b: fail (misleading) 2026-10-17 — The first sentence incorrectly describes the manipulation needed to reach the standard polar form; it suggests dividing by a constant in the denominator, which is not the correct step. The rest of the solution is fine, but the misleading statement could confuse a student about how to rewrite the given equation.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly divide the numerator and denominator by 7 to match the standard form r = ed/(1 + e cos θ), leading to an incorrect directrix calculation. The numerator should be 8/7, not 8, so d = 8/49, not 8/7.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the eccentricity and conic type by normalizing the denominator. The directrix calculation is correct: for the form r = ed/(1 + e cos theta), the directrix is x = d.
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.