Conics in polar form
Problem 8.514 · medium
For the conic \( \displaystyle r = \frac{6}{2 - 4 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{3}{1 - 2 \sin{\left(\theta \right)}} = \frac{6}{2 - 4 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-4}\right|}{2} = 2 \]e = 2: > 1, a hyperbola.✓ Proved
- \[ \frac{6}{2 \cdot 2} = \frac{3}{2} \]ed = 3, so d = 3/2.✓ Proved
Answer \( e = 2,\ \text{hyperbola},\ \text{directrix } y = -3/2 \)
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: fail (error) — The first sentence incorrectly states the standard polar form for a conic. It claims the denominator should involve θ via ασω (cos θ) and that the form is r = ed/(1 ± e cos θ), whereas the given equation involves sin θ. The correct standard form for a vertical conic is r = ed/(1 ± e sin θ). This misstatement misleads the reader about the proper conversion and the orientation of the conic.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The directrix is identified as y = -3/2, but the standard form r = ed/(1 - e sin theta) corresponds to a directrix at y = -d. Since d = 3/2, the directrix should be y = -3/2. Wait, let me re-evaluate. The standard form is r = ep / (1 - e sin theta) for directrix y = -p. Here we have r = 3 / (1 - 2 sin theta). So ep = 3, e=2 => p = 3/2. The directrix is y = -p = -3/2. The stated answer says y = -3/2. The solution calculates d = 3/2. It does not explicitly state the sign of the directrix in the text, but the final answer in the problem statement is y = -3/2. Let's look at the solution steps again. Step 4 says 'ed = 3, so d = 3/2'. It does not explicitly link d to the directrix equation y = -d or y = d. However, usually 'd' represents the distance from the focus to the directrix. The position depends on the sign in the denominator. 1 - e sin(theta) implies the directrix is below the pole (y = -d). The solution fails to explicitly state that the directrix is y = -3/2, only calculating the distance d. But the 'Stated answer' is separate from the 'Solution' lines. The solution lines end at step 4. Step 4 concludes d=3/2. It does not conclude the directrix equation. Is this an error? The problem asks to 'find the directrix'. The solution finds 'd'. It misses the final step of writing the equation of the directrix. This is an incomplete solution. However, looking at the provided 'Stated answer', it matches the correct math. The review is of the 'Solution' lines. The solution lines do not produce the final answer 'y = -3/2'. They stop at 'd = 3/2'. This is a failure to answer the full question. Also, step 1 says 'Divide top and bottom by the constant... to get r = ed/(1 +/- e cos theta)'. The problem has sin, not cos. This is a misleading statement in step 1.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] The directrix is identified as y = -3/2, but the standard form r = ed/(1 - e sin theta) corresponds to a directrix at y = -d. Since d = 3/2, the directrix should be y = -3/2. Wait, let me re-evaluate. The standard form is r = ep / (1 - e sin theta) for directrix y = -p. Here we have r = 3 / (1 - 2 sin theta). So ep = 3, e=2 => p = 3/2. The directrix is y = -p = -3/2. The stated answer says y = -3/2. The solution calculates d = 3/2. It does not explicitly state the sign of the directrix in the text, but the final answer in the problem statement is y = -3/2. Let's look at the solution steps again. Step 4 says 'ed = 3, so d = 3/2'. It does not explicitly link d to the directrix equation y = -d or y = d. However, usually 'd' represents the distance from the focus to the directrix. The position depends on the sign in the denominator. 1 - e sin(theta) implies the directrix is below the pole (y = -d). The solution fails to explicitly state that the directrix is y = -3/2, only calculating the distance d. But the 'Stated answer' is separate from the 'Solution' lines. The solution lines end at step 4. Step 4 concludes d=3/2. It does not conclude the directrix equation. Is this an error? The problem asks to 'find the directrix'. The solution finds 'd'. It misses the final step of writing the equation of the directrix. This is an incomplete solution. However, looking at the provided 'Stated answer', it matches the correct math. The review is of the 'Solution' lines. The solution lines do not produce the final answer 'y = -3/2'. They stop at 'd = 3/2'. This is a failure to answer the full question. Also, step 1 says 'Divide top and bottom by the constant... to get r = ed/(1 +/- e cos theta)'. The problem has sin, not cos. This is a misleading statement in step 1.gpt-oss:20b: fail (error) 2026-10-19 — The first sentence incorrectly states the standard polar form for a conic. It claims the denominator should involve θ via ασω (cos θ) and that the form is r = ed/(1 ± e cos θ), whereas the given equation involves sin θ. The correct standard form for a vertical conic is r = ed/(1 ± e sin θ). This misstatement misleads the reader about the proper conversion and the orientation of the conic.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the directrix as y = -3/2. For the form r = ed / (1 - e sin θ), the directrix is y = -d. Since d = 3/2, the directrix should be y = -3/2. Wait, let me re-evaluate. The standard form is r = ed / (1 - e sin θ). Here we have r = 3 / (1 - 2 sin θ). So ed = 3 and e = 2, which implies d = 3/2. The term is -e sin θ, which corresponds to a directrix of y = -d. So y = -3/2. The stated answer says y = -3/2. The solution steps don't explicitly state the directrix equation, but step 4 calculates d. The prompt asks to review the solution lines. Line 4 calculates d=3/2. It does not state the directrix equation. However, the 'Stated answer' is part of the problem description, not the solution lines to be reviewed? No, usually the solution leads to the answer. The solution stops at calculating d. It does not explicitly write the directrix equation. But more importantly, let's look at the sign. The denominator is 1 - 2 sin θ. This matches the form 1 - e sin θ. The directrix for 1 - e sin θ is y = -d. So y = -3/2. The stated answer is correct. The solution steps are algebraically correct. Is there a missing step? The solution calculates d but doesn't write the final directrix equation. This might be considered incomplete or 'style'. But let's look closer. Step 1 says 'Divide top and bottom... to get r = ed/(1 ± e cos θ)'. It mentions cos, but the problem has sin. This is a misleading statement in the setup sentence. It should say sin. This is a 'misleading' or 'error' in the sentence description. The sentence claims the target form involves cos, but the problem involves sin. This is a factual error in the sentence.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to account for the sign of the sine term. The form r = ed / (1 - e sin θ) corresponds to a directrix at y = -d, not y = d. Since e=2 and ed=3 implies d=3/2, the directrix is y = -3/2, but the derivation in step 4 does not explicitly link the negative sign in the denominator to the negative y-coordinate, risking confusion about the sign convention.gpt-oss:20b: fail (error) 2026-10-18 — Sentence 1 incorrectly states the standard form as r=ed/(1±e cosθ) and omits the necessary division by the constant 2; the correct form involves sinθ, not cosθ, and the sign must be handled properly.
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.