Conics in polar form
Problem 8.506 · medium
For the conic \( \displaystyle r = \frac{3}{6 \sin{\left(\theta \right)} + 3} \), 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).Reviewed
- \[ \frac{1}{2 \sin{\left(\theta \right)} + 1} = \frac{3}{6 \sin{\left(\theta \right)} + 3} \]The standard form.✓ Proved
- \[ \frac{\left|{6}\right|}{3} = 2 \]e = 2: > 1, a hyperbola.✓ Proved
- \[ \frac{3}{2 \cdot 3} = \frac{1}{2} \]ed = 1, so d = 1/2.✓ Proved
Answer \( e = 2,\ \text{hyperbola},\ \text{directrix } y = 1/2 \)
Lines: 3 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the eccentricity and conic type. The directrix is correctly identified as y = 1/2 based on the standard form r = ed / (1 + e sin theta), where the directrix is y = d.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the eccentricity and conic type. The directrix is correctly identified as y = 1/2 based on the standard form r = ed / (1 + e sin theta), where the directrix is y = d.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution incorrectly identifies the directrix as y = 1/2. Since the standard form involves sin(theta), the directrix is horizontal (y = d), but the sign in the denominator is positive (+ e sin theta), which corresponds to a directrix below the pole (y = -d). Thus, the directrix is y = -1/2, not y = 1/2.gpt-oss:20b: fail (error) 2026-10-18 — The solution misapplies the standard polar form. Dividing by 3 gives r = 1/(2 sinθ + 1), which is not of the form ed/(1 ± e cosθ); the equation in step 2 is algebraically incorrect, and the identification of the directrix is not justified.qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution incorrectly identifies the directrix as y = 1/2. Since the standard form involves sin(theta), the directrix is horizontal (y = d), but the sign in the denominator is positive (+3), which corresponds to the form r = ed/(1 + e sin theta). This implies the directrix is above the pole, so y = +1/2 is actually correct for the distance, but the standard form derivation in step 1 claims '1 +/- e cos theta' while using sin. More critically, the standard form r = ed / (1 + e sin theta) has directrix y = d. Here ed = 1 and e = 2, so d = 1/2. The directrix is y = 1/2. Wait, let me re-evaluate. Standard forms: 1. r = ed / (1 + e cos theta) -> Directrix x = d 2. r = ed / (1 - e cos theta) -> Directrix x = -d 3. r = ed / (1 + e sin theta) -> Directrix y = d 4. r = ed / (1 - e sin theta) -> Directrix y = -d The given equation is r = 3 / (3 + 6 sin theta) = 1 / (1 + 2 sin theta). So e = 2, ed = 1 => d = 1/2. The form is 1 + e sin theta, so the directrix is y = d = 1/2. The stated answer says directrix y = 1/2. This is correct. Let's look at the sentences. Sentence 1: "Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos theta)." This sentence is misleading/wrong because it specifies cos theta, but the problem has sin theta. It should say "cos theta or sin theta". However, is this an 'error' or 'misleading'? It's a bad generalization. Let's look closer at the prompt's request. "find what is wrong". Is the final answer correct? Yes. Is the reasoning sound? Step 1 says get r = ed/(1 +/- e cos theta). This is factually incorrect as a general statement for all conics, as they can involve sine. Step 2 shows the algebra for sine. Step 3 calculates e. Step 4 calculates d. The error is in Sentence 1. It restricts the form to cosine, which is false. A student might think sine forms don't exist or are invalid. This is a 'misleading' or 'error' in the setup description. Given the severity options, 'error' fits best because the statement "get r = ... cos theta" is mathematically false as a universal rule for converting conic equations. It fails to model the problem correctly by ignoring the sine case in the general formula description.gpt-oss:20b: fail (error) 2026-10-18 — The solution misstates the standard form (uses cos θ instead of sin θ) and the algebra in step 2 is incorrect, so the setup and conclusions are not justified.
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.