Conics in polar form
Problem 8.455 · medium
For the conic \( \displaystyle r = \frac{6}{8 \sin{\left(\theta \right)} + 2} \), 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).
- \[ \frac{3}{4 \sin{\left(\theta \right)} + 1} = \frac{6}{8 \sin{\left(\theta \right)} + 2} \]The standard form.✓ Proved
- \[ \frac{\left|{8}\right|}{2} = 4 \]e = 4: > 1, a hyperbola.✓ Proved
- \[ \frac{6}{2 \cdot 4} = \frac{3}{4} \]ed = 3, so d = 3/4.✓ Proved
Answer \( e = 4,\ \text{hyperbola},\ \text{directrix } y = 3/4 \)
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 (error) — The solution misapplies the standard polar form. The equation r=6/(8 sinθ+2) is not immediately in the form r=ed/(1±e cosθ); dividing by 2 only rewrites the denominator as 4 sinθ+1, but the standard form requires a cosine term (or a shift of the angle). The step claiming that e=|8|/2=4 follows directly from the coefficient of sinθ is not justified without first converting the expression to the correct standard form. Consequently the setup and the derivation of e, the type of conic, and the directrix are not rigorously supported.qwen3.6:27b-mlx: pass — The solution correctly transforms the equation into standard polar form, identifies the eccentricity and conic type, and correctly determines the directrix equation based on the sine term.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly transforms the equation into standard polar form, identifies the eccentricity and conic type, and correctly determines the directrix equation based on the sine term.gpt-oss:20b: fail (error) 2026-10-19 — The solution misapplies the standard polar form. The equation r=6/(8 sinθ+2) is not immediately in the form r=ed/(1±e cosθ); dividing by 2 only rewrites the denominator as 4 sinθ+1, but the standard form requires a cosine term (or a shift of the angle). The step claiming that e=|8|/2=4 follows directly from the coefficient of sinθ is not justified without first converting the expression to the correct standard form. Consequently the setup and the derivation of e, the type of conic, and the directrix are not rigorously supported.qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution incorrectly identifies the directrix as y = 3/4. Since the standard form involves sin(theta), the directrix is horizontal (y = d), but the sign in the denominator (positive) implies the directrix is above the pole, so y = 3/4 is technically correct for the distance, but the standard form derivation usually yields r = ed/(1 + e sin theta) for directrix y = +d. However, the main error is in step 1's claim: dividing by 2 gives r = 3/(4 sin theta + 1), so e=4 and ed=3, implying d=3/4. The directrix is y = 3/4. Wait, let me re-read. r = 6/(2 + 8 sin theta) = 3/(1 + 4 sin theta). Standard form r = ed/(1 + e sin theta). So e=4, ed=3 => d=3/4. Directrix is y = 3/4. The solution says 'directrix y = 3/4'. This seems correct. Let me check the sign. 1 + e sin theta corresponds to directrix y = +d. Yes. Is there a subtle error? Step 1 says 'Divide top and bottom by the constant in the denominator'. The constant is 2. 6/2 = 3. 8/2 = 4. So r = 3/(1 + 4 sin theta). This matches r = ed/(1 + e sin theta) with e=4, ed=3. d=3/4. Directrix y = 3/4. The solution seems correct. Why would it fail? Ah, look at step 2. It writes `(6/2)/((8/2)*sin(theta) + 1)`. This is `3/(4 sin theta + 1)`. This is correct. Step 3 calculates `Abs(8)/2 = 4`. This is e. Correct. Step 4 calculates `(6/2)/4 = 3/4`. This is d. Correct. The directrix is y = 3/4. The stated answer is `e = 4, hyperbola, directrix y = 3/4`. The solution derives this. Is there a convention issue? Sometimes directrix is written as y = -d if the sign is negative. Here sign is positive. So y = +3/4. It seems correct. Let me look closer at the prompt's request. 'find what is wrong'. Maybe the issue is that the directrix is y = 3/4, but the problem might imply a different orientation? No, sin theta means horizontal directrix. Is it possible the directrix is y = -3/4? If r = ed/(1 - e sin theta), directrix is y = -d. Here we have +. So y = +d. It seems correct. Let me check the 'constant in the denominator' phrasing. The denominator is 8 sin theta + 2. The constant term is 2. Dividing by 2 makes the constant term 1. This is the standard procedure. I don't see an error. Wait, is the directrix y = 3/4 or y = -3/4? For r = ep/(1 + e sin theta), the directrix is y = p. Here p = d = 3/4. So y = 3/4. Correct. Is it possible the solution is 'pass'? Let me double check if I missed a 'fail' condition. 'Adversarially'. Is there a subtle mistake? Step 1: 'Divide top and bottom by the constant in the denominator'. This is a valid method. Step 2: The equation is proved. Step 3: e=4. Step 4: d=3/4. Conclusion: directrix y=3/4. Everything seems logically sound. However, often in these problems, if the form is r = ed/(1 + e sin theta), the directrix is y = d. If the form is r = ed/(1 - e sin theta), the directrix is y = -d. Here we have +. So y = 3/4. I will mark it as pass.gpt-oss:20b: fail (misleading) 2026-10-16 — The first sentence claims the standard form is r = ed/(1 ± e cosθ) after dividing by 2, but the given equation involves sinθ, so the correct standard form is r = ed/(1 ± e sinθ). This misstatement could confuse a student about the orientation of the conic.qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as y = 3/4. Since the standard form involves sin(theta), the directrix is horizontal, but the posigpt-oss:20b: fail (error) 2026-10-15 — The first sentence incorrectly states the standard form as r = ed/(1 ± e cos θ) even though the given equation involves sin θ. The correct standard form should use sin, e.g., r = ed/(1 ± e sin θ). This misidentification of the form leads to confusion in the subsequent steps.
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.