Conics in polar form
Problem 8.446 · medium
For the conic \( \displaystyle r = \frac{10}{1 - 6 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{10}{1 \left(- \frac{6 \sin{\left(\theta \right)}}{1} + 1\right)} = \frac{10}{1 - 6 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-6}\right|}{1} = 6 \]e = 6: > 1, a hyperbola.✓ Proved
- \[ \frac{10}{1 \cdot 6} = \frac{5}{3} \]ed = 10, so d = 5/3.✓ Proved
Answer \( e = 6,\ \text{hyperbola},\ \text{directrix } y = -5/3 \)
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 the standard form as r = ed/(1 ± e cos θ); the given equation involves sin θ, so the correct form is r = ed/(1 ± e sin θ).qwen3.6:27b-mlx: fail (error) — The solution correctly identifies the eccentricity and the distance d, but fails to determine the correct sign and orientation of the directrix. For the form r = ed / (1 - e sin(theta)), the directrix is y = -d, not y = d. The stated answer y = -5/3 is correct, but the derivation in step 4 only yields the magnitude d = 5/3 without explaining why the directrix is at y = -5/3 rather than y = 5/3. A student following this logic might incorrectly conclude the directrix is y = 5/3.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution correctly identifies the eccentricity and the distance d, but fails to determine the correct sign and orientation of the directrix. For the form r = ed / (1 - e sin(theta)), the directrix is y = -d, not y = d. The stated answer y = -5/3 is correct, but the derivation in step 4 only yields the magnitude d = 5/3 without explaining why the directrix is at y = -5/3 rather than y = 5/3. A student following this logic might incorrectly conclude the directrix is y = 5/3.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly states the standard form as r = ed/(1 ± e cos θ); the given equation involves sin θ, so the correct form is r = ed/(1 ± e sin θ).qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to determine the sign of the directrix. For the form r = ed / (1 - e sin(theta)), the directrix is y = -d, not y = d. The correct directrix is y = -5/3, but the derivation in step 4 implies d = 5/3 without addressing the sign convention required by the minus sign in the denominator.gpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly claims the standard form is r = ed/(1 ± e cos θ) after dividing by the constant, but the given equation involves sin θ, not cos θ. This misidentifies the orientation of the directrix and the form of the conic, leading to an incorrect setup.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution correctly identifies the eccentricity and type but fails to determine the correct sign for the directrix. For the form r = ed / (1 - e sin(theta)), the directrix is y = -d, so the directrix should be y = -5/3, not y = 5/3 (or the solution implies d=5/3 without specifying the line equation correctly relative to the sign in the denominator).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 2026-10-18 with SymPy 1.14.0.