Conics in polar form
Problem 8.319 · medium
For the conic \( \displaystyle r = \frac{5}{6 \sin{\left(\theta \right)} + 2} \), 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).
- \[ \frac{5}{2 \left(3 \sin{\left(\theta \right)} + 1\right)} = \frac{5}{6 \sin{\left(\theta \right)} + 2} \]The standard form.✓ Proved
- \[ \frac{\left|{6}\right|}{2} = 3 \]e = 3: > 1, a hyperbola.✓ Proved
- \[ \frac{5}{2 \cdot 3} = \frac{5}{6} \]ed = 5/2, so d = 5/6.✓ Proved
Answer \( e = 3,\ \text{hyperbola},\ \text{directrix } y = 5/6 \)
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 (misleading) — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ) even though the given equation involves sinθ. This misleads a student about the correct form for a conic with a vertical directrix. The subsequent algebraic steps are fine, but the initial statement is inaccurate.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), not vertical. While the final answer happens to state y = 5/6, the derivation steps (3 and 4) only calculate the value d without justifying the axis, leaving the student without the rule that sin implies horizontal and cos implies vertical.
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), not vertical. While the final answer happens to state y = 5/6, the derivation steps (3 and 4) only calculate the value d without justifying the axis, leaving the student without the rule that sin implies horizontal and cos implies vertical.gpt-oss:20b: fail (misleading) 2026-10-17 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ) even though the given equation involves sinθ. This misleads a student about the correct form for a conic with a vertical directrix. The subsequent algebraic steps are fine, but the initial statement is inaccurate.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to specify the orientation of the directrix. Since the equation involves 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). Additionally, the sign of the directrix is ambiguous without checking the quadrant or standard form conventions (usually y = d for +sin, y = -d for -sin), though y=5/6 is likely correct for +sin, the reasoning is incomplete.gpt-oss:20b: fail (misleading) 2026-10-15 — 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 into using the wrong form for the directrix and focus. The rest of the algebra is correct, but the initial statement is misleading.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution correctly identifies the eccentricity and conic type, but fails to specify the orientation of the directrix. Since the equation involves sin(theta), the directrix is horizontal (y = 5/6), not vertical. While the final stated answer is correct, the derivation in step 4 does not justify why the directrix is y = d rather than x = d, which is a critical part of 'finding the directrix'.gpt-oss:20b: pass 2026-10-14
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.