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Conics in polar form

Problem 8.432 · medium

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

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

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 (style) — [domain objection, downgraded to style] The solution correctly identifies e=2 and d=5, but fails to determine the correct sign for the directrix. The form r = ed/(1 - e cos theta) corresponds to a directrix at x = -d, so the directrix is x = -5. While the stated answer is correct, the solution text does not derive the sign, leaving a gap in the logic regarding the position of the directrix relative to the pole.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution correctly identifies e=2 and d=5, but fails to determine the correct sign for the directrix. The form r = ed/(1 - e cos theta) corresponds to a directrix at x = -d, so the directrix is x = -5. While the stated answer is correct, the solution text does not derive the sign, leaving a gap in the logic regarding the position of the directrix relative to the pole.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution correctly identifies e=2 and d=5, but fails to determine the correct sign for the directrix. The form r = ed/(1 - e cos theta) corresponds to a directrix at x = -d, so the directrix is x = -5. While the stated answer is correct, the solution steps do not justify the negative sign, leaving a gap in the logic that a student might miss.
gpt-oss:20b: fail (misleading) 2026-10-16 — The solution incorrectly concludes the directrix is x = -5. For the polar equation r = ed/(1 - e cosθ) with e = 2 and d = 5, the directrix is the line x = +5, not x = -5. The sentence claiming x = -5 misleads a student about the relationship between the sign in the denominator and the location of the directrix.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to account for the sign in the denominator. The form r = ed / (1 - e cos θ) corresponds to a directrix at x = -d, not x = d. Since e=2 and ed=10 implies d=5, the directrix is x = -5. While the final answer happens to be correct, the reasoning in step 4 (implying d=5 directly determines the directrix without addressing the sign convention) is flawed and would lead to errors in other cases (e.g., if the denominator were 1 + 2 cos θ). A rigorous solution must explicitly link the minus sign in the denominator to the negative x-coordinate of the directrix.
gpt-oss:20b: pass 2026-10-15

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.