Conics in polar form
Problem 8.435 · medium
For the conic \( \displaystyle r = \frac{12}{5 - 4 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{12}{5 \left(1 - \frac{4 \sin{\left(\theta \right)}}{5}\right)} = \frac{12}{5 - 4 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-4}\right|}{5} = \frac{4}{5} \]e = 4/5: < 1, an ellipse.✓ Proved
- \[ \frac{12}{5 \cdot \frac{4}{5}} = 3 \]ed = 12/5, so d = 3.✓ Proved
Answer \( e = \frac{4}{5},\ \text{ellipse},\ \text{directrix } y = -3 \)
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 solution misidentifies the standard form (uses cos instead of sin) and incorrectly concludes the directrix is y = –3; for r = ed/(1 – e sinθ) with d = 3 the directrix is y = 3.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution correctly identifies the eccentricity and the distance d, but fails to determine the correct orientation of the directrix. Since the te
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution correctly identifies the eccentricity and the distance d, but fails to determine the correct orientation of the directrix. Since the tegpt-oss:20b: fail (error) 2026-10-18 — The solution misidentifies the standard form (uses cos instead of sin) and incorrectly concludes the directrix is y = –3; for r = ed/(1 – e sinθ) with d = 3 the directrix is y = 3.qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution fails to determine the correct sign and position of the directrix. The term -4 sin(theta) indicates the directrix is horizontal and below the pole (y = -d), but the solution only calculates the magnitude d=3 without explicitly linking the negative sign in the denominator to the negative y-coordinate, resulting in an incomplete derivation for the directrix equation.gpt-oss:20b: fail (error) 2026-10-16 — The solution misidentifies the standard polar form (it writes cos instead of sin) and incorrectly determines the directrix. For r=12/(5-4sinθ)= (12/5)/(1-(4/5)sinθ), the directrix is y=+3, not y=-3. The conclusion about the directrix is therefore wrong.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The directrix is identified as y = -3, but the standard form r = ed/(1 - e sin theta) corresponds to a directrix at y = -d. Since d=3, the directrix is indeed y = -3. However, the solution fails to explicitly justify the sign of the directrix based on the minus sign in the denominator (1 - e sin theta implies y = -d). While the final answer is correct, the reasoning in step 4 simply calculates d=3 without linking the sign of the sine term to the position of the directrix, which is a critical part of 'finding' the directrix. More importantly, step 1 claims to divide to get r = ed/(1 ± e cos theta), but the problem involves sin(theta). This is a factual error in the stated method.gpt-oss:20b: fail (misleading) 2026-10-15 — The solution incorrectly identifies the directrix as y = –3. For r = ed/(1 – e sinθ) the directrix is y = d (positive), so with d = 3 the correct directrix is y = 3. Additionally, the text mistakenly refers to a cos‑term and the sign in the denominator is wrong, which would mislead a student.
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.