Conics in polar form
Problem 8.507 · medium
For the conic \( \displaystyle r = \frac{10}{8 \sin{\left(\theta \right)} + 3} \), 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).
- \[ \frac{10}{3 \left(\frac{8 \sin{\left(\theta \right)}}{3} + 1\right)} = \frac{10}{8 \sin{\left(\theta \right)} + 3} \]The standard form.✓ Proved
- \[ \frac{\left|{8}\right|}{3} = \frac{8}{3} \]e = 8/3: > 1, a hyperbola.✓ Proved
- \[ \frac{10}{3 \cdot \frac{8}{3}} = \frac{5}{4} \]ed = 10/3, so d = 5/4.✓ Proved
Answer \( e = \frac{8}{3},\ \text{hyperbola},\ \text{directrix } y = 5/4 \)
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 the standard form r=ed/(1±e cosθ) after dividing by 3, but the denominator contains sinθ, not cosθ. Consequently the identification of e and the directrix is not justified by the given expression. The solution misapplies the polar conic formula and therefore is incorrect.qwen3.6:27b-mlx: fail (error) — The solution fails to identify the directrix as a horizontal line (y = 5/4) because the denominator contains sin(theta), not cos(theta). The standard form with sin(theta) corresponds to a directrix of the form y = d, but the setup sentence incorrectly cites the cosine form, leading to an incomplete and potentially confusing derivation 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 = 5/4) because the denominator contains sin(theta), not cos(theta). The standard form with sin(theta) corresponds to a directrix of the form y = d, but the setup sentence incorrectly cites the cosine form, leading to an incomplete and potentially confusing derivation of the directrix's orientation.gpt-oss:20b: fail (error) 2026-10-19 — The first sentence incorrectly claims the standard form r=ed/(1±e cosθ) after dividing by 3, but the denominator contains sinθ, not cosθ. Consequently the identification of e and the directrix is not justified by the given expression. The solution misapplies the polar conic formula and therefore is incorrect.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 = 5/4 rather than y = -5/4 or x = 5/4. While the final answer happens to be correct for the standard form r = ed/(1+esin(theta)), the reasoning is incomplete and potentially misleading regarding the sign and orientation.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly claims that dividing by the constant in the denominator yields the standard polar form r = ed/(1 ± e cosθ). The given conic involves sinθ, not cosθ, and the resulting expression (10/3)/((8/3)sinθ+1) is not in the required form. Thus the setup is incorrect and the subsequent conclusions are invalid.qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly transforms the equation into standard polar form, identifies the eccentricity and conic type, and correctly determines the directrix based on the sine term indicating a horizontal directrix y = d.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly states the standard polar form: it uses cos θ instead of sin θ for a denominator containing sin θ. This misidentifies the orientation of the directrix and could mislead a student about the correct form of the equation.
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.