Conics in polar form
Problem 8.460 · medium
For the conic \( \displaystyle r = \frac{6}{2 \sin{\left(\theta \right)} + 5} \), 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).
- \[ \frac{6}{5 \left(\frac{2 \sin{\left(\theta \right)}}{5} + 1\right)} = \frac{6}{2 \sin{\left(\theta \right)} + 5} \]The standard form.✓ Proved
- \[ \frac{\left|{2}\right|}{5} = \frac{2}{5} \]e = 2/5: < 1, an ellipse.✓ Proved
- \[ \frac{6}{5 \cdot \frac{2}{5}} = 3 \]ed = 6/5, so d = 3.✓ Proved
Answer \( e = \frac{2}{5},\ \text{ellipse},\ \text{directrix } y = 3 \)
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 (error) — The first sentence incorrectly states that the equation can be written in the form r = ed/(1 ± e cosθ). The given polar equation involves sinθ, not cosθ, so the standard form used is r = ed/(1 + e sinθ). This misidentification of the standard form could mislead a student.qwen3.6:27b-mlx: fail (error) — The solution correctly identifies the eccentricity and distance d, but fails to specify the orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = d or y = -d), not vertical. The stated answer 'y = 3' is correct, but the derivation in step 4 merely calculates d=3 without linking the sine term to the y-axis, leaving the student without the rule for determining the directrix's orientation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution correctly identifies the eccentricity and distance d, but fails to specify the orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = d or y = -d), not vertical. The stated answer 'y = 3' is correct, but the derivation in step 4 merely calculates d=3 without linking the sine term to the y-axis, leaving the student without the rule for determining the directrix's orientation.gpt-oss:20b: fail (error) 2026-10-19 — The first sentence incorrectly states that the equation can be written in the form r = ed/(1 ± e cosθ). The given polar equation involves sinθ, not cosθ, so the standard form used is r = ed/(1 + e sinθ). This misidentification of the standard form could mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to identify the directrix as a horizontal line (y = 3) because the standard form involves sin(theta), not cos(theta). The text in step 1 incorrectly cites the cosine form, leading to a missing justification for the orientation of the directrix.gpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly claims that dividing by the constant yields the standard polar form r=ed/(1±e cosθ). The given denominator contains sinθ, not cosθ, so the expression is not in the required form. Consequently, the subsequent identification of e and d is not justified by the stated method.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: pass 2026-10-15
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.