Conics in polar form
Problem 8.505 · medium
For the conic \( \displaystyle r = \frac{11}{3 \sin{\left(\theta \right)} + 2} \), 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).
- \[ \frac{11}{2 \left(\frac{3 \sin{\left(\theta \right)}}{2} + 1\right)} = \frac{11}{3 \sin{\left(\theta \right)} + 2} \]The standard form.✓ Proved
- \[ \frac{\left|{3}\right|}{2} = \frac{3}{2} \]e = 3/2: > 1, a hyperbola.✓ Proved
- \[ \frac{11}{2 \cdot \frac{3}{2}} = \frac{11}{3} \]ed = 11/2, so d = 11/3.✓ Proved
Answer \( e = \frac{3}{2},\ \text{hyperbola},\ \text{directrix } y = 11/3 \)
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 (misleading) — The first sentence incorrectly states the standard form as involving γ (cos θ) when the given equation uses sin θ. The rest of the solution correctly identifies e=3/2, the hyperbola, and the directrix y=11/3.qwen3.6:27b-mlx: fail (error) — The solution fails to identify the directrix as a horizontal line (y = 11/3) because the denominator contains sin(theta), not cos(theta). It incorrectly implies the standard form is r = ed/(1 ± e cos theta) while using sine, leading to a missing specification of the directrix's orientation.
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 = 11/3) because the denominator contains sin(theta), not cos(theta). It incorrectly implies the standard form is r = ed/(1 ± e cos theta) while using sine, leading to a missing specification of the directrix's orientation.gpt-oss:20b: fail (misleading) 2026-10-19 — The first sentence incorrectly states the standard form as involving γ (cos θ) when the given equation uses sin θ. The rest of the solution correctly identifies e=3/2, the hyperbola, and the directrix y=11/3.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to identify the orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = d), but the solution does not explicitly state this or justify why it is y rather than x, leading to an incomplete identification of the directrix.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly states that dividing by the constant in the denominator yields the standard polar form r = ed/(1 ± e cosθ). The given denominator contains sinθ, not cosθ, so the resulting expression does not match the standard conic form. Consequently, the subsequent identification of e and the directrix is based on a faulty setup.qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution identifies the directrix as y = 11/3, but fails to specify the sign. Since the denominator contains +sin(theta), the directrix is y = +d, so y = 11/3 is technically correct in value, but the reasoning in step 1 implies a generic form without addressing the sign convention for the directrix location relative to the pole. More critically, the standard form derivation is slightly loose: dividing by 2 gives r = (11/2) / (1 + (3/2)sin(theta)). Here ed = 11/2 and e = 3/2, so d = 11/3. The directrix for r = ed / (1 + e sin theta) is y = d. The solution states 'directrix y = 11/3' which is correct. However, looking closely at step 1, it says 'Divide... to get r = ed/(1 +/- e cos theta)'. The problem has sin, not cos. This is a mismatch in the stated method vs the actual function. The student might be confused why they used sin in the equation but cited a cos formula. This is misleading.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly states that the standard polar form is r = ed/(1 ± e cos θ) when the given equation involves sin θ. The correct form should involve sin θ, and the division step should be clarified accordingly.
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.