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Conics in polar form

Problem 8.546 · medium

For the conic \( \displaystyle r = \frac{2}{1 - 7 \cos{\left(\theta \right)}} \), 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).
\[ \frac{2}{1 \left(- \frac{7 \cos{\left(\theta \right)}}{1} + 1\right)} = \frac{2}{1 - 7 \cos{\left(\theta \right)}} \]
The standard form.✓ Proved
\[ \frac{\left|{-7}\right|}{1} = 7 \]
e = 7: > 1, a hyperbola.✓ Proved
\[ \frac{2}{1 \cdot 7} = \frac{2}{7} \]
ed = 2, so d = 2/7.✓ Proved

Answer \( e = 7,\ \text{hyperbola},\ \text{directrix } x = -2/7 \)

Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 (error) — The solution incorrectly identifies the directrix. For the polar equation r=ed/(1−e cosθ) with e>1, the directrix is x=d, not x=−d. Thus the directrix should be x=2/7, not x=−2/7.
qwen3.6:27b-mlx: fail (error) — The directrix is incorrectly identified as x = -2/7. For the form r = ed / (1 - e cos(theta)), the directrix is x = -d, which is x = -2/7. Wait, let me re-verify. Standard form r = ed / (1 - e cos theta) corresponds to directrix x = -d. Here e=7, ed=2 => d=2/7. So directrix is x = -2/7. The stated answer says x = -2/7. The solution calculates d=2/7 but does not explicitly state the sign convention for the directrix relative to the denominator term. However, usually, r = ep / (1 - e cos theta) implies the directrix is x = -p. Let's check the standard derivation. r - r e cos theta = ep => r = ep / (1 - e cos theta). In Cartesian: sqrt(x^2+y^2) = e(x+p) if directrix is x=-p? No. Distance to focus (origin) is r. Distance to directrix x = -d is |x+d|. r = e|x+d|. If x > -d, r = e(x+d) => r = ex + ed => r - ex = ed => r = ed / (1 - e cos theta). So directrix is x = -d. Here d = 2/7, so x = -2/7. The stated answer is correct. The solution steps are: 1. Setup. 2. Identify form. 3. Identify e. 4. Identify d. It fails to explicitly connect d to the directrix equation x = -d. It just says 'd = 2/7'. A student might think the directrix is x = 2/7 or y = 2/7. The sentence 'ed = 2, so d = 2/7' is an algebraic step, but the final identification of the directrix is missing from the sentences provided in the solution block, or rather, the solution block ends at calculating d. The 'Stated answer' is separate. The solution provided is just steps 1-4. Step 4 calculates d. It does not state the directrix. The prompt asks to review the 'Solution'. The solution does not conclude with the directrix equation. It stops at d=2/7. This is incomplete. Also, step 1 says 'Divide top and bottom...'. The equation in step 2 is just an identity check, not the result of division. The original equation is already in the form 2/(1-7cos). No division was needed to get it into standard form, it was already there. Step 1 is misleading/wrong about the action taken. Step 2 just confirms the equation is the same. The solution is incomplete and the setup description is inaccurate.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-05 — The directrix is incorrectly identified as x = -2/7. For the form r = ed / (1 - e cos(theta)), the directrix is x = -d, which is x = -2/7. Wait, let me re-verify. Standard form r = ed / (1 - e cos theta) corresponds to directrix x = -d. Here e=7, ed=2 => d=2/7. So directrix is x = -2/7. The stated answer says x = -2/7. The solution calculates d=2/7 but does not explicitly state the sign convention for the directrix relative to the denominator term. However, usually, r = ep / (1 - e cos theta) implies the directrix is x = -p. Let's check the standard derivation. r - r e cos theta = ep => r = ep / (1 - e cos theta). In Cartesian: sqrt(x^2+y^2) = e(x+p) if directrix is x=-p? No. Distance to focus (origin) is r. Distance to directrix x = -d is |x+d|. r = e|x+d|. If x > -d, r = e(x+d) => r = ex + ed => r - ex = ed => r = ed / (1 - e cos theta). So directrix is x = -d. Here d = 2/7, so x = -2/7. The stated answer is correct. The solution steps are: 1. Setup. 2. Identify form. 3. Identify e. 4. Identify d. It fails to explicitly connect d to the directrix equation x = -d. It just says 'd = 2/7'. A student might think the directrix is x = 2/7 or y = 2/7. The sentence 'ed = 2, so d = 2/7' is an algebraic step, but the final identification of the directrix is missing from the sentences provided in the solution block, or rather, the solution block ends at calculating d. The 'Stated answer' is separate. The solution provided is just steps 1-4. Step 4 calculates d. It does not state the directrix. The prompt asks to review the 'Solution'. The solution does not conclude with the directrix equation. It stops at d=2/7. This is incomplete. Also, step 1 says 'Divide top and bottom...'. The equation in step 2 is just an identity check, not the result of division. The original equation is already in the form 2/(1-7cos). No division was needed to get it into standard form, it was already there. Step 1 is misleading/wrong about the action taken. Step 2 just confirms the equation is the same. The solution is incomplete and the setup description is inaccurate.
gpt-oss:20b: fail (error) 2026-11-05 — The solution incorrectly identifies the directrix. For the polar equation r=ed/(1−e cosθ) with e>1, the directrix is x=d, not x=−d. Thus the directrix should be x=2/7, not x=−2/7.
qwen3.6:27b-mlx: inconclusive 2026-11-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is identified as x = -2/7, but for the form r = ed/(1 - e cos(theta)), the directrix is x = -d. Since d = 2/7, the directrix should be
gpt-oss:20b: fail (misleading) 2026-11-04 — The first sentence incorrectly claims that dividing by the constant in the denominator yields the standard form; in fact the denominator’s constant is 1, so no division is needed. The statement also ambiguously uses ± instead of the correct 1−e cosθ form, potentially confusing the reader about the sign convention.
qwen3.6:27b-mlx: inconclusive 2026-11-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is incorrectly identified as x = -2/7. For the form r = ed/(1 - e cos θ), the directrix is x = -d, which yields x = -2/7. Wait, let me
gpt-oss:20b: fail (error) 2026-11-04 — The solution incorrectly identifies the directrix. The given polar equation r=2/(1-7cosθ) is in the form r=ed/(1-e cosθ) with e=7 and ed=2, so d=2/7. For this form the directrix is the vertical line x=+d, not x=−2/7. The first sentence also misstates the manipulation needed to reach the standard form.

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.