Conics in polar form
Problem 8.314 · medium
For the conic \( \displaystyle r = \frac{10}{4 \sin{\left(\theta \right)} + 3} \), 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).
- \[ \frac{10}{3 \left(\frac{4 \sin{\left(\theta \right)}}{3} + 1\right)} = \frac{10}{4 \sin{\left(\theta \right)} + 3} \]The standard form.✓ Proved
- \[ \frac{\left|{4}\right|}{3} = \frac{4}{3} \]e = 4/3: > 1, a hyperbola.✓ Proved
- \[ \frac{10}{3 \cdot \frac{4}{3}} = \frac{5}{2} \]ed = 10/3, so d = 5/2.✓ Proved
Answer \( e = \frac{4}{3},\ \text{hyperbola},\ \text{directrix } y = 5/2 \)
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: fail (error) — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cos θ) when the given equation involves sin θ. It should be r = ed/(1 ± e sin θ).qwen3.6:27b-mlx: fail (error) — The solution fails to specify 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 distinguish it from a vertical directrix (x = d) which would result from cos(theta). Without this, the answer 'y = 5/2' is not justified by the steps shown.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to specify 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 distinguish it from a vertical directrix (x = d) which would result from cos(theta). Without this, the answer 'y = 5/2' is not justified by the steps shown.gpt-oss:20b: fail (error) 2026-10-17 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cos θ) when the given equation involves sin θ. It should be r = ed/(1 ± e sin θ).qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution fails to identify the directrix as horizontal (y = 5/2) because the denominator contains sin(theta), not cos(theta). It also omits the sign check required to determine if the directrix is above or below the pole.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence incorrectly assumes the standard polar form r=ed/(1±e cosθ) applies after dividing by the constant. The given denominator contains sinθ, so the correct form is r=ed/(1±e sinθ). This misidentification of the standard form leads to an incorrect identification of the directrix and the conic type.qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: fail (misleading) 2026-10-14 — The first sentence incorrectly states the standard polar form as r=ed/(1±e cosθ) when the given equation involves sinθ; this could mislead a student about the correct orientation of the conic.
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 2026-10-17 with SymPy 1.14.0.