Conics in polar form
Problem 8.512 · medium
For the conic \( \displaystyle r = \frac{9}{3 - 2 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{3}{1 - \frac{2 \sin{\left(\theta \right)}}{3}} = \frac{9}{3 - 2 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-2}\right|}{3} = \frac{2}{3} \]e = 2/3: < 1, an ellipse.✓ Proved
- \[ \frac{9}{3 \cdot \frac{2}{3}} = \frac{9}{2} \]ed = 3, so d = 9/2.✓ Proved
Answer \( e = \frac{2}{3},\ \text{ellipse},\ \text{directrix } y = -9/2 \)
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: fail (style) — [domain objection, downgraded to style] The directrix is identified as y = -9/2, but the standard form r = ed/(1 - e sin θ) corresponds to a directrix at y = -d. Since d = 9/2, the directrix is indeed y = -9/2. Wait, let me re-evaluate. The standard form is r = ed / (1 - e sin θ). Here we have r = 3 / (1 - (2/3) sin θ). So ed = 3 and e = 2/3. Thus d = 3 / (2/3) = 9/2. The form 1 - e sin θ implies the directrix is y = -d. So y = -9/2. The solution seems correct. Let me double check the sign convention. r = ep / (1 - e sin θ) -> directrix y = -p. r = ep / (1 + e sin θ) -> directrix y = p. The equation is r = 9 / (3 - 2 sin θ) = 3 / (1 - (2/3) sin θ). This matches 1 - e sin θ. So directrix is y = -d = -9/2. The stated answer is correct. Why did I think it was wrong? Ah, I need to check if the problem asks for *the* directrix. An ellipse has two directrices. Usually, the one corresponding to the focus at the pole is the one derived from the polar equation. The other directrix is not given by this single polar equation directly without transformation. The question asks for 'the directrix', implying the one associated with the focus at the origin. The solution finds d=9/2 and states y=-9/2. This is correct. Is there any subtle error? Step 1 says 'Divide top and bottom by the constant... to get r = ed/(1 ± e cos θ)'. It mentions cos θ but the problem has sin θ. This is a minor style issue or a typo in the sentence, but the subsequent steps use sin correctly. Step 3 calculates e = 2/3. Step 4 calculates d = 9/2. The final answer combines these. The sentence in step 1 is slightly imprecise by mentioning cos instead of sin, but it's a generic template. Is it 'misleading'? It might confuse a student into thinking it's a cosine form. However, the math follows the sine form. Let's look closer. Step 1: 'Divide top and bottom... to get r = ed/(1 ± e cos θ)'. This is a generic statement. The actual equation has sin. The student might be misled to look for cos. But the next lines use sin. I will mark it as style/misleading because the template mentioned cos while the problem is sin. Actually, looking at the provided solution text again: '1. [sentence] Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ).' This is technically incorrect for this specific problem which uses sin. It should say 'cos or sin'. This is a style/clarity issue. But wait, is the verdict 'pass'? The math is correct. The final answer is correct. The sentence is just a bit sloppy. I'll call it 'style' if it's just sloppy, or 'misleading' if it teaches a false rule. It teaches that the form is always cos, which is false. So 'misleading'. But the verdict is 'fail' only if it's wrong. Is it wrong? The method works. The result is correct. I will say 'pass' because the core logic is sound and the result is correct, and the 'cos' mention is likely a generic placeholder that the student corrected in their head or in the next step. Actually, strict adherence: 'Check that... nothing would teach a student something false.' Teaching that the standard form is always with cos is false. So 'fail', 'misleading'. Let's look at the options. 'error' is for wrong steps. 'misleading' is for false lessons. I'll go with misleading.
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 = -9/2, but the standard form r = ed/(1 - e sin θ) corresponds to a directrix at y = -d. Since d = 9/2, the directrix is indeed y = -9/2. Wait, let me re-evaluate. The standard form is r = ed / (1 - e sin θ). Here we have r = 3 / (1 - (2/3) sin θ). So ed = 3 and e = 2/3. Thus d = 3 / (2/3) = 9/2. The form 1 - e sin θ implies the directrix is y = -d. So y = -9/2. The solution seems correct. Let me double check the sign convention. r = ep / (1 - e sin θ) -> directrix y = -p. r = ep / (1 + e sin θ) -> directrix y = p. The equation is r = 9 / (3 - 2 sin θ) = 3 / (1 - (2/3) sin θ). This matches 1 - e sin θ. So directrix is y = -d = -9/2. The stated answer is correct. Why did I think it was wrong? Ah, I need to check if the problem asks for *the* directrix. An ellipse has two directrices. Usually, the one corresponding to the focus at the pole is the one derived from the polar equation. The other directrix is not given by this single polar equation directly without transformation. The question asks for 'the directrix', implying the one associated with the focus at the origin. The solution finds d=9/2 and states y=-9/2. This is correct. Is there any subtle error? Step 1 says 'Divide top and bottom by the constant... to get r = ed/(1 ± e cos θ)'. It mentions cos θ but the problem has sin θ. This is a minor style issue or a typo in the sentence, but the subsequent steps use sin correctly. Step 3 calculates e = 2/3. Step 4 calculates d = 9/2. The final answer combines these. The sentence in step 1 is slightly imprecise by mentioning cos instead of sin, but it's a generic template. Is it 'misleading'? It might confuse a student into thinking it's a cosine form. However, the math follows the sine form. Let's look closer. Step 1: 'Divide top and bottom... to get r = ed/(1 ± e cos θ)'. This is a generic statement. The actual equation has sin. The student might be misled to look for cos. But the next lines use sin. I will mark it as style/misleading because the template mentioned cos while the problem is sin. Actually, looking at the provided solution text again: '1. [sentence] Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ).' This is technically incorrect for this specific problem which uses sin. It should say 'cos or sin'. This is a style/clarity issue. But wait, is the verdict 'pass'? The math is correct. The final answer is correct. The sentence is just a bit sloppy. I'll call it 'style' if it's just sloppy, or 'misleading' if it teaches a false rule. It teaches that the form is always cos, which is false. So 'misleading'. But the verdict is 'fail' only if it's wrong. Is it wrong? The method works. The result is correct. I will say 'pass' because the core logic is sound and the result is correct, and the 'cos' mention is likely a generic placeholder that the student corrected in their head or in the next step. Actually, strict adherence: 'Check that... nothing would teach a student something false.' Teaching that the standard form is always with cos is false. So 'fail', 'misleading'. Let's look at the options. 'error' is for wrong steps. 'misleading' is for false lessons. I'll go with misleading.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution fails to determine the correct sign and orientation of the directrix. The term -2 sin(theta) indicates the directrix is horizontal (y = k) and below the pole (y = -d), but the final answer y = -9/2 is derived without explicitly stating this geometric reasoning, and the intermediate step 'ed = 3' is ambiguous regarding the sign of d relative to the sine term.gpt-oss:20b: fail (error) 2026-10-18 — Sentence 1 incorrectly states the standard form as r = ed/(1 ± e cosθ) and claims to divide by the constant in the denominator to obtain that form. The given conic uses sinθ, so the correct standard form is r = ed/(1 ± e sinθ).qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution correctly identifies the eccentricity and the type of conic, but fails to determine the correct sign for the directrix. The standard form with a negative sine term, r = ed/(1 - e sin θ), corresponds to a directrix at y = -d. Since d = 9/2, the directrix is y = -9/2. While the final answer listed in the problem statement is correct, the solution steps do not explicitly justify the negative sign for the directrix based on the minus sign in the denominator, potentially misleading a student into thinking the directrix is always y = d or y = -d regardless of the sign in the equation. Specifically, step 4 calculates d but does not link the sign of the term in the denominator to the position of the directrix.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly states the standard form and uses cos instead of sin; the directrix sign is also wrong – it should be y=+9/2, not –9/2.
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.