Conics in polar form
Problem 8.508 · medium
For the conic \( \displaystyle r = \frac{12}{4 - 6 \cos{\left(\theta \right)}} \), 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).
- \[ \frac{3}{1 - \frac{3 \cos{\left(\theta \right)}}{2}} = \frac{12}{4 - 6 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-6}\right|}{4} = \frac{3}{2} \]e = 3/2: > 1, a hyperbola.✓ Proved
- \[ \frac{12}{4 \cdot \frac{3}{2}} = 2 \]ed = 3, so d = 2.✓ Proved
Answer \( e = \frac{3}{2},\ \text{hyperbola},\ \text{directrix } x = -2 \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is identified as x = -2, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = -d. Since d = 2, the directrix i
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is identified as x = -2, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = -d. Since d = 2, the directrix igpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The directrix is incorrectly identified as x = -2. Since the term is -6 cos(theta), the standard form is 1 - e cos(theta), which corresponds to a dirgpt-oss:20b: fail (error) 2026-10-18 — The solution incorrectly identifies the directrix as x = –2. For r = ed/(1 – e cosθ) the directrix is x = +d, so with d = 2 the correct directrix is x = 2, not –2.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the directrix as x = -2. Since the term in the denominator is -6 cos(theta), the standard form is r = ed / (1 - e cos(theta)), which corresponds to a directrix at x = -d. With d=2, the directrix is x = -2. Wait, let me re-evaluate. Standard form r = ed / (1 + e cos theta) has directrix x = d. r = ed / (1 - e cos theta) has directrix x = -d. Here we have 1 - (3/2)cos(theta), so directrix is x = -2. The stated answer says x = -2. The solution calculates d=2 but does not explicitly state the sign logic for the directrix location, merely stating 'directrix x = -2' in the final answer block which is correct. However, looking at step 4, it says 'ed = 3, so d = 2'. It does not explicitly link the sign of the cosine term to the position of the directrix in the text, but the final answer provided in the prompt is x = -2. Let's look closer at the prompt's 'Stated answer'. It says 'directrix x = -2'. The solution steps end at calculating d=2. The final answer is separate. Does the solution *text* contain an error? Step 1 says 'Divide... to get r = ed/(1 +/- e cos theta)'. Step 2 shows the algebra. Step 3 finds e. Step 4 finds d. The solution text does not explicitly state the directrix equation, but the 'Stated answer' does. The review is of the 'Solution' lines. The solution lines do not conclude with the directrix equation. They just find d. Is finding d sufficient? The problem asks to 'find the directrix'. The solution stops at finding d. It fails to state the directrix equation in the solution steps. However, usually 'find the directrix' implies finding the line equation. The solution calculates d=2. It misses the step of determining the sign/location. But wait, looking at the provided 'Stated answer', it is part of the problem description or the expected output? The prompt says 'Stated answer: ...'. The 'Solution' is lines 1-4. Line 4 ends with 'd=2'. It does not state the directrix. This is an incomplete solution. But is it an 'error'? It fails to answer the full question. Also, check the sign. r = 12 / (4 - 6 cos theta) = 3 / (1 - 1.5 cos theta). e = 1.5. ed = 3 => d = 2. Form is 1 - e cos theta. Directrix is x = -d = -2. The stated answer is correct. The solution steps are correct algebraically but incomplete in deriving the final geometric object. However, often in these formats, if the final answer is provided separately, the steps just need to support it. But step 4 says 'ed=3, so d=2'. It doesn't mention the directrix. I will mark it as error because it doesn't actually find the directrix, just the distance d, and doesn't specify the line equation in the steps.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution correctly finds e=3/2 and d=2, but the directrix should be x=+2, not x=−2. The sign of the cosine term in the polar equation determines the side of the directrix relative to the focus.
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 2026-10-19 with SymPy 1.14.0.