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Conics in polar form

Problem 8.551 · medium

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

Answer \( e = \frac{4}{3},\ \text{hyperbola},\ \text{directrix } x = -1/4 \)

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 solution claims the directrix is x = -1/4, but for the form r = ed/(1 - e cosθ) the directrix is x = +d = 1/4. The sign of the directrix is incorrect, giving a false result for a student.
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The directrix is identified as x = -1/4, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = -d. Since d = 1/4, the directrix should be x = -1/4. Wait, let me re-evaluate. The form is 1 - e cos(theta). This usually implies the directrix is x = -d? No. Standard forms: r = ed/(1 + e cos theta) -> x = d. r = ed/(1 - e cos theta) -> x = -d. Here we have 1 - (4/3)cos(theta). So e=4/3, d=1/4. The directrix is x = -1/4. The stated answer says x = -1/4. The solution calculates d=1/4 but does not explicitly state the sign/direction logic for the directrix, just calculating d. However, looking closely at step 2: (1/3)/((-4/3)*cos(theta) + 1). This is 1/(3-4cos(theta)). The denominator is 1 - (4/3)cos(theta). This matches the form 1 - e cos(theta). For this form, the directrix is x = -d. So x = -1/4. The stated answer is correct. The solution steps are algebraically correct. Is there an error? Step 3 says Abs(-4)/3 = 4/3. This is correct for e. Step 4 calculates d. The solution doesn't explicitly write the final directrix equation, but the stated answer does. The prompt asks to review the solution lines. The sentences are sparse. Sentence 1 is a method. The equations are proved. The conclusion in the stated answer is correct. Is there a subtle error? Let's check the directrix sign convention again. r = ep / (1 - e cos theta). Directrix is x = -p. Here p = d = 1/4. So x = -1/4. The stated answer is correct. The solution is sound. Why would it fail? Maybe the sentence 'Divide top and bottom by the constant...' is slightly imprecise? No, it's standard. Let's look at the 'Abs(-4)/3' step. e is defined as positive. The coefficient of cos is -4/3. So e = |-4/3| = 4/3. This is correct. The form is 1 - e cos theta. This corresponds to directrix x = -d. The solution finds d=1/4. It does not explicitly state the directrix equation in the steps, but the stated answer has it. The review is of the solution lines. The lines are correct. Verdict: pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-11-05 — [domain objection, downgraded to style] The directrix is identified as x = -1/4, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = -d. Since d = 1/4, the directrix should be x = -1/4. Wait, let me re-evaluate. The form is 1 - e cos(theta). This usually implies the directrix is x = -d? No. Standard forms: r = ed/(1 + e cos theta) -> x = d. r = ed/(1 - e cos theta) -> x = -d. Here we have 1 - (4/3)cos(theta). So e=4/3, d=1/4. The directrix is x = -1/4. The stated answer says x = -1/4. The solution calculates d=1/4 but does not explicitly state the sign/direction logic for the directrix, just calculating d. However, looking closely at step 2: (1/3)/((-4/3)*cos(theta) + 1). This is 1/(3-4cos(theta)). The denominator is 1 - (4/3)cos(theta). This matches the form 1 - e cos(theta). For this form, the directrix is x = -d. So x = -1/4. The stated answer is correct. The solution steps are algebraically correct. Is there an error? Step 3 says Abs(-4)/3 = 4/3. This is correct for e. Step 4 calculates d. The solution doesn't explicitly write the final directrix equation, but the stated answer does. The prompt asks to review the solution lines. The sentences are sparse. Sentence 1 is a method. The equations are proved. The conclusion in the stated answer is correct. Is there a subtle error? Let's check the directrix sign convention again. r = ep / (1 - e cos theta). Directrix is x = -p. Here p = d = 1/4. So x = -1/4. The stated answer is correct. The solution is sound. Why would it fail? Maybe the sentence 'Divide top and bottom by the constant...' is slightly imprecise? No, it's standard. Let's look at the 'Abs(-4)/3' step. e is defined as positive. The coefficient of cos is -4/3. So e = |-4/3| = 4/3. This is correct. The form is 1 - e cos theta. This corresponds to directrix x = -d. The solution finds d=1/4. It does not explicitly state the directrix equation in the steps, but the stated answer has it. The review is of the solution lines. The lines are correct. Verdict: pass.
gpt-oss:20b: fail (misleading) 2026-11-05 — The solution claims the directrix is x = -1/4, but for the form r = ed/(1 - e cosθ) the directrix is x = +d = 1/4. The sign of the directrix is incorrect, giving a false result for a student.
qwen3.6:27b-mlx: inconclusive 2026-11-04 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The directrix is incorrect. For the form r = ed / (1 - e cos θ), the directrix is x = -d. Here d = 1/4, so the directrix is x = -1/4. However, the st
gpt-oss:20b: pass 2026-11-04
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 = -1/4, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = -d (or x = d depending on conve
gpt-oss:20b: fail (error) 2026-11-04 — The solution incorrectly identifies the directrix as x=−1/4. For the standard form r=ed/(1−e cosθ) the directrix is x=+d=1/4, not −1/4. The earlier steps are correct, but the conclusion about the directrix is wrong.

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.