Conics in polar form
Problem 8.549 · medium
For the conic \( \displaystyle r = \frac{12}{7 \sin{\left(\theta \right)} + 1} \), 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).
- \[ \frac{12}{1 \left(\frac{7 \sin{\left(\theta \right)}}{1} + 1\right)} = \frac{12}{7 \sin{\left(\theta \right)} + 1} \]The standard form.✓ Proved
- \[ \frac{\left|{7}\right|}{1} = 7 \]e = 7: > 1, a hyperbola.✓ Proved
- \[ \frac{12}{1 \cdot 7} = \frac{12}{7} \]ed = 12, so d = 12/7.✓ Proved
Answer \( e = 7,\ \text{hyperbola},\ \text{directrix } y = 12/7 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 polar form of a conic. It claims r = ed/(1 ± e cosθ) for a denominator containing sinθ, which is inconsistent. The correct form for a conic with a vertical directrix is r = ed/(1 ± e sinθ). Because of this misstatement the subsequent identification of the conic is not justified by the given reasoning.qwen3.6:27b-mlx: fail (error) — The solution fails to identify the directrix as a horizontal line (y = 12/7) because the denominator contains sin(theta), not cos(theta). It also incorrectly claims the standard form has a plus sign in the denominator (1 + e sin theta) while the problem has 7 sin theta + 1, which is consistent, but the critical error is omitting the variable y for the directrix, stating just '12/7' which is ambiguous or incorrect as a line equation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-05 — The solution fails to identify the directrix as a horizontal line (y = 12/7) because the denominator contains sin(theta), not cos(theta). It also incorrectly claims the standard form has a plus sign in the denominator (1 + e sin theta) while the problem has 7 sin theta + 1, which is consistent, but the critical error is omitting the variable y for the directrix, stating just '12/7' which is ambiguous or incorrect as a line equation.gpt-oss:20b: fail (error) 2026-11-05 — The first sentence incorrectly states the standard polar form of a conic. It claims r = ed/(1 ± e cosθ) for a denominator containing sinθ, which is inconsistent. The correct form for a conic with a vertical directrix is r = ed/(1 ± e sinθ). Because of this misstatement the subsequent identification of the conic is not justified by the given reasoning.qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution fails to specify the orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = d), not vertical. The stated answer y = 12/7 is correct, but the derivation in step 1 references the cosine form without adapting it for sine, and step 4 does not explicitly link the sine term to the y-coordinate, making the logic incomplete and potentially misleading regarding the relationship between the trig function and the directrix orientation.gpt-oss:20b: fail (error) 2026-11-04 — The first sentence incorrectly claims that dividing by the constant in the denominator yields the standard form r=ed/(1±e cosθ). The given equation already has a sinθ term, so the correct standard form is r=ed/(1±e sinθ). This misstatement could mislead a student about the appropriate form for a horizontal directrix.qwen3.6:27b-mlx: fail (error) 2026-11-04 — 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 justify why it is y rather than x, leading to an incomplete identification of the directrix.gpt-oss:20b: fail (error) 2026-11-04 — The first sentence incorrectly assumes that dividing by the constant in the denominator yields the standard form r=ed/(1±e cosθ). The given equation has a sinθ term, not cosθ, and the constant in the denominator is 1, so no division changes the form. Consequently the subsequent identification of e, the type of conic, and the directrix are not justified by the stated reasoning.
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 2026-11-05 with SymPy 1.14.0.