Conics in polar form
Problem 8.511 · medium
For the conic \( \displaystyle r = \frac{11}{4 \sin{\left(\theta \right)} + 2} \), 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).
- \[ \frac{11}{2 \left(2 \sin{\left(\theta \right)} + 1\right)} = \frac{11}{4 \sin{\left(\theta \right)} + 2} \]The standard form.✓ Proved
- \[ \frac{\left|{4}\right|}{2} = 2 \]e = 2: > 1, a hyperbola.✓ Proved
- \[ \frac{11}{2 \cdot 2} = \frac{11}{4} \]ed = 11/2, so d = 11/4.✓ Proved
Answer \( e = 2,\ \text{hyperbola},\ \text{directrix } y = 11/4 \)
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 first sentence incorrectly claims that dividing by the constant in the denominator yields the standard form r = ed/(1 ± e cosθ). The given conic involves sinθ, so the correct standard form is r = ed/(1 ± e sinθ). The rest of the solution is based on this incorrect statement, leading to a misleading justification of the form and the identification of the directrix.qwen3.6:27b-mlx: fail (error) — The solution fails to specify the orientation of the directrix. Since the equation involves sin(theta), the directrix is horizontal (y = 11/4), but the derivation steps do not explicitly link the sine term to the y-axis orientation, leaving the student without the crucial logic for determining the directrix's equation form.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to specify the orientation of the directrix. Since the equation involves sin(theta), the directrix is horizontal (y = 11/4), but the derivation steps do not explicitly link the sine term to the y-axis orientation, leaving the student without the crucial logic for determining the directrix's equation form.gpt-oss:20b: fail (error) 2026-10-19 — The first sentence incorrectly claims that dividing by the constant in the denominator yields the standard form r = ed/(1 ± e cosθ). The given conic involves sinθ, so the correct standard form is r = ed/(1 ± e sinθ). The rest of the solution is based on this incorrect statement, leading to a misleading justification of the form and the identification of the directrix.qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the eccentricity and conic type. Although the directrix is stated as y = 11/4, this is consistent with the standard form r = ed/(1 + e sin θ) derived in the steps, which corresponds to a directrix at y = d.gpt-oss:20b: fail (error) 2026-10-18 — The solution incorrectly assumes the polar equation can be written in the standard form r=ed/(1±e cosθ) by merely dividing by the constant in the denominator. The given equation involves sinθ, not cosθ, and the conversion to standard form requires a rotation of the coordinate system, which is not addressed. Thus the setup is wrong and the conclusions about e, the type of conic, and the directrix are not justified.qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly manipulates the equation into standard polar form, identifies the eccentricity and conic type, and correctly determines the directrix location and orientation.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence incorrectly claims the standard form is r=ed/(1±e cosθ); the given equation involves sinθ, so the form should be r=ed/(1±e sinθ). This misstatement could mislead a student about the correct polar representation of conics.
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.