Conics in polar form
Problem 8.456 · medium
For the conic \( \displaystyle r = \frac{8}{\sin{\left(\theta \right)} + 5} \), 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).
- \[ \frac{8}{5 \left(\frac{\sin{\left(\theta \right)}}{5} + 1\right)} = \frac{8}{\sin{\left(\theta \right)} + 5} \]The standard form.✓ Proved
- \[ \frac{\left|{1}\right|}{5} = \frac{1}{5} \]e = 1/5: < 1, an ellipse.✓ Proved
- \[ \frac{8}{5 \cdot 1 \cdot \frac{1}{5}} = 8 \]ed = 8/5, so d = 8.✓ Proved
Answer \( e = \frac{1}{5},\ \text{ellipse},\ \text{directrix } y = 8 \)
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 form for a polar conic; it should involve sin θ, not cos θ, and the sign must match the given denominator. The conclusion that the directrix is y = 8 is also wrong—the correct directrix for r = ed/(1+e sinθ) with d = 8 is y = –8.qwen3.6:27b-mlx: fail (error) — The solution fails to identify the directrix as a horizontal line (y = 8) because the standard form uses sin(theta), not cos(theta). The setup in step 1 is incomplete/misleading as it does not distinguish between the vertical directrix (cos) and horizontal directrix (sin) cases.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to identify the directrix as a horizontal line (y = 8) because the standard form uses sin(theta), not cos(theta). The setup in step 1 is incomplete/misleading as it does not distinguish between the vertical directrix (cos) and horizontal directrix (sin) cases.gpt-oss:20b: fail (error) 2026-10-19 — The first sentence incorrectly states the standard form for a polar conic; it should involve sin θ, not cos θ, and the sign must match the given denominator. The conclusion that the directrix is y = 8 is also wrong—the correct directrix for r = ed/(1+e sinθ) with d = 8 is y = –8.qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to identify the correct directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = d), but the sign igpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cos θ) when the given equation contains sin θ. The correct form should involve sin θ (directrix y = d). This misstatement misleads the reader about the proper identification of the conic’s orientation.qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution incorrectly identifies the directrix as y = 8. Since the denominator contains sin(theta), the directrix is horizontal, but the standard form r = ed / (1 + e sin theta) corresponds to a directrix at y = +d. However, the original equation is r = 8 / (sin theta + 5). Rewriting this as r = (8/5) / (1 + (1/5)sin theta) gives e = 1/5 and ed = 8/5, so d = 8. The directrix is y = 8. Wait, let me re-evaluate. The standard form is r = ed / (1 + e sin theta). Here we have r = (8/5) / (1 + (1/5)sin theta). So e = 1/5, ed = 8/5 => d = 8. The directrix is y = d = 8. The solution says y = 8. Is this correct? Yes. But wait, the prompt says 'find the directrix'. The solution concludes 'directrix y = 8'. Let's check the sign. The form is 1 + e sin theta. This corresponds to a directrix above the pole, y = d. So y = 8 is correct. Why did I think it was an error? Let me re-read the solution steps. Step 1 says 'Divide top and bottom... to get r = ed/(1 +/- e cos theta)'. It mentions cos theta, but the problem has sin theta. This is a misleading statement in the setup sentence, but the subsequent algebra uses sin. Step 2 shows the correct algebraic manipulation. Step 3 identifies e. Step 4 calculates d. The final answer is not explicitly stated as a sentence, but implied by the steps. The question asks to review the sentences. Sentence 1 says 'get r = ed/(1 +/- e cos theta)'. This is technically incorrect because the problem involves sin theta, not cos theta. While the student likely meant 'trig function', stating 'cos' specifically is misleading/wrong for this specific problem. However, the algebra in step 2 correctly uses sin. Is this an error or style? It's a 'misleading' sentence because it claims the target form has cos, which is false for this problem. But does it affect the result? No. The result is correct. Let's look closer. Is the directrix y=8 correct? r = 8/(sin theta + 5). At theta = pi/2, r = 8/6 = 4/3. y = r sin theta = 4/3. At theta = 3pi/2, r = 8/4 = 2. y = r sin theta = -2. The center of the ellipse is on the y-axis. The directrix for r = ed/(1+e sin theta) is y=d. Here d=8. So y=8. The solution is correct. The only flaw is Sentence 1 mentioning 'cos' instead of 'sin' or 'trig function'. This is a 'style' or 'misleading' issue. Given the strict adversarial role, claiming the standard form involves cos when it involves sin is a factual error in the sentence, even if the math follows correctly. I will mark it as misleading.gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence incorrectly states the standard form as r=ed/(1±e cosθ) even though the given equation involves sinθ. It also omits the ± sign and the correct identification of the directrix orientation. This misleads a student about the proper polar form for a conic with a sinθ term.
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.