Conics in polar form
Problem 8.445 · medium
For the conic \( \displaystyle r = \frac{8}{2 \sin{\left(\theta \right)} + 5} \), 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).
- \[ \frac{8}{5 \left(\frac{2 \sin{\left(\theta \right)}}{5} + 1\right)} = \frac{8}{2 \sin{\left(\theta \right)} + 5} \]The standard form.✓ Proved
- \[ \frac{\left|{2}\right|}{5} = \frac{2}{5} \]e = 2/5: < 1, an ellipse.✓ Proved
- \[ \frac{8}{5 \cdot \frac{2}{5}} = 4 \]ed = 8/5, so d = 4.✓ Proved
Answer \( e = \frac{2}{5},\ \text{ellipse},\ \text{directrix } y = 4 \)
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 solution misapplies the standard polar form: it keeps a σ term instead of converting to π, incorrectly identifies the eccentricity from the coefficient of σ, and derives the directrix as y=4 when the correct directrix for r=ed/(1+eσ) is y=-d.qwen3.6:27b-mlx: pass — The solution correctly transforms the polar equation into standard form, identifies the eccentricity and conic type, and calculates the directrix distance. Since the denominator involves sin(theta), the directrix is horizontal (y = d), which matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly transforms the polar equation into standard form, identifies the eccentricity and conic type, and calculates the directrix distance. Since the denominator involves sin(theta), the directrix is horizontal (y = d), which matches the stated answer.gpt-oss:20b: fail (error) 2026-10-18 — The solution misapplies the standard polar form: it keeps a σ term instead of converting to π, incorrectly identifies the eccentricity from the coefficient of σ, and derives the directrix as y=4 when the correct directrix for r=ed/(1+eσ) is y=-d.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly transforms the polar equation into standard form, identifies the eccentricity and conic type, and accurately calculates the directrix distance and orientation based on the sine term.gpt-oss:20b: fail (error) 2026-10-16 — Sentence 1 incorrectly states that dividing by the constant yields the standard form r=ed/(1±e cosθ). The given equation involves sinθ, not cosθ, and the division does not produce the ±cosθ form. The subsequent steps rely on this incorrect statement, so the setup is flawed.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly manipulates the polar equation into standard form, accurately identifies the eccentricity and conic type, and correctly determines the directrix equation based on the sine term.gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence incorrectly states the standard form as r = ed/(1 ± e cosθ) when the denominator contains sinθ; it should be r = ed/(1 ± e sinθ). This misleads a student about the correct polar equation form for a conic with a vertical directrix.
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 2026-10-18 with SymPy 1.14.0.