Conics in polar form
Problem 8.550 · medium
For the conic \( \displaystyle r = \frac{6}{1 - \sin{\left(\theta \right)}} \), 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).
- \[ \frac{6}{1 \left(- \frac{\sin{\left(\theta \right)}}{1} + 1\right)} = \frac{6}{1 - \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-1}\right|}{1} = 1 \]e = 1: = 1, a parabola.✓ Proved
- \[ \frac{6}{1 \cdot 1} = 6 \]ed = 6, so d = 6.✓ Proved
Answer \( e = 1,\ \text{parabola},\ \text{directrix } y = -6 \)
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 (error) — The first sentence incorrectly claims the standard form involves cos θ and that one should divide by the constant in the denominator; the given polar equation is already in the correct form r=ed/(1−e sinθ). Moreover, the directrix for r=6/(1−sinθ) is y=6, not y=−6, so the conclusion about the directrix is wrong.qwen3.6:27b-mlx: fail (error) — The solution correctly identifies e=1 and d=6, but fails to determine the correct position and orientation of the directrix. For r = ed/(1 - sin(theta)), the directrix is y = -d, so y = -6. However, the solution does not explicitly state this derivation or justify why the directrix is y = -6 rather than y = 6 or x = +/- 6, leaving a critical gap in the reasoning for the final answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-05 — The solution correctly identifies e=1 and d=6, but fails to determine the correct position and orientation of the directrix. For r = ed/(1 - sin(theta)), the directrix is y = -d, so y = -6. However, the solution does not explicitly state this derivation or justify why the directrix is y = -6 rather than y = 6 or x = +/- 6, leaving a critical gap in the reasoning for the final answer.gpt-oss:20b: fail (error) 2026-11-05 — The first sentence incorrectly claims the standard form involves cos θ and that one should divide by the constant in the denominator; the given polar equation is already in the correct form r=ed/(1−e sinθ). Moreover, the directrix for r=6/(1−sinθ) is y=6, not y=−6, so the conclusion about the directrix is wrong.qwen3.6:27b-mlx: fail (style) 2026-11-04 — [domain objection, downgraded to style] The solution correctly identifies e=1 and d=6, but fails to determine the correct orientation of the directrix. The term -sin(theta) implies the directrix is horizontal and below the pole (y = -d), but the solution does not explicitly state this geometric deduction, leaving the sign of the directrix ambiguous or derived implicitly without justification.gpt-oss:20b: fail (error) 2026-11-04 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ); the given equation involves sinθ, not cosθ, so the setup is wrong. The subsequent algebraic steps are fine, but the initial model is incorrect.qwen3.6:27b-mlx: fail (error) 2026-11-04 — The solution fails to account for the sign in the denominator. The form r = ed/(1 - sin θ) corresponds to a directrix of y = -d. Since d=6, the directrix is y = -6, but the reasoning in step 1 implies a generic form without specifying how the minus sign maps to the directrix location, which is the critical part of the problem. More importantly, standard convention usually writes the denominator as 1 + e sin(theta) or 1 - e sin(theta). Here, comparing 6/(1 - sin(theta)) to ed/(1 - e sin(theta)) gives e=1 and ed=6 => d=6. The directrix for 1 - e sin(theta) is y = -d. The stated answer says y = -6, which is correct. However, the sentence in step 1 is vague ('Divide top and bottom...') and doesn't explicitly state the relationship between the sign and the directrix position, which is where students typically fail. But looking closer at the provided 'Stated answer', it is correct. Let's re-read the prompt's constraints. 'Check that... each conclusion follows from the lines before it'. Step 3 concludes e=1. Step 4 concludes d=6. The final answer is not explicitly derived in the sentences, but implied. The main issue is that the sentence in step 1 is just a procedural instruction, not a logical derivation. But is it wrong? No. Is the setup correct? Yes. Is the conclusion correct? Yes. Wait, let me look at the 'Stated answer' vs the 'Solution'. The solution stops at finding d=6. It does not explicitly state the directrix equation in the steps provided. The prompt asks to review the *solution*. The solution provides e, identifies the conic, and finds d. It does not explicitly write 'directrix y = -6'. However, usually 'find the directrix' implies finding the equation. The solution finds d=6. A student might think the directrix is y=6 or x=6. The sign is crucial. The form is 1 - sin(theta). This corresponds to y = -d. The solution does not state this mapping. Therefore, the solution is incomplete/misleading because it doesn't connect d=6 to the specific directrix equation y=-6, leaving the sign ambiguous for a learner.gpt-oss:20b: fail (error) 2026-11-04 — The solution incorrectly identifies the directrix as y = –6. For the standard form r = ed/(1 – e sinθ) with e = 1 and d = 6, the directrix is y = +6, not –6. The other steps are correct.
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.